\documentclass[10pt]{book} % Must use LaTeX 2e \usepackage{html,makeidx,color,colordvi,graphics,graphicx,amssymb,amsmath} \global\emergencystretch = .9\hsize \makeindex % % changes for latex2html : % % Sierpi\'nski -> Sierpinski % ............. % % \v Cech -> Cech % % D\c ebski Debski % % Wa\.zewski Wazewski % % \v Stanko % Podbrdsk\'y % % Ma\'ckowiak % % Jel\'\i nek % % Ar\'evalo % % % % % % % % % % % % \begin{document} % The start of the document \title{Examples in Continuum Theory} \author{Janusz J. Charatonik, Pawel Krupski and Pavel Pyrih} \date{ \today} \maketitle \tableofcontents %\renewcommand{\thechapter}{\Roman{chapter}} %\listoffigures % \newcommand{\FIGURE}[5][]% { \setcounter{figure}{\value{subsection}} \addtocounter{figure}{-1} \begin{figure} \begin{makeimage} \end{makeimage} \begin{latexonly} \begin{center} \includegraphics[#1]{#3#4.eps} \end{center} \end{latexonly} \begin{htmlonly} \htmladdimg[center]{../#3#4.gif} \end{htmlonly} \caption{ ( #4 ) #5} \label{#2} \end{figure} } \newcommand{\GIF}[5][]% { \setcounter{figure}{\value{subsection}} \addtocounter{figure}{-1} \begin{htmlonly} \begin{figure} \begin{makeimage} \end{makeimage} \htmladdimg[center]{../#3#4.gif} \caption{ ( #4 ) #5} \label{#2} \end{figure} \end{htmlonly} } \newcommand{\newtargetlink}[2]{% \htmladdnormallink{#1}{#2" target="_new}} % % \newcommand{\STORY}[1]% { \begin{htmlonly} \smallskip Here you can find \newtargetlink{source files}{../#1} of this example. \smallskip Here you can check the \newtargetlink{table}{http://www.karlin.mff.cuni.cz/~pyrih/e/e/s/a/table.xls} of properties of individual continua. \smallskip Here you can \newtargetlink{ read Notes}{http://www.karlin.mff.cuni.cz/~pyrih/cctbook.html} or \newtargetlink{ write to Notes}{http://www.karlin.mff.cuni.cz/~pyrih/addcct.html} ies of individual continua. \end{htmlonly} } % \STORY{../s/c0003/s0020/e0050} % \GIF[scale=1]{}{../s/c0003/s0020/e0050/}{A}{buckethandle} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % HTML REFERENCES AND INDEX ENTRIES % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % \g = GO SOMEWHERE % % usage : % \g{continuum} % output : % continuum (underlined in html version) % \def \g #1{{\htmlref{#1}{#1}}} % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % \gs = GO SPECIAL % % usage : % \gs {Sierpi\'nski Universal Plane Curve}{Sierpinski Universal Plane Curve} % % output : % Sierpi\'nski Universal Plane Curve (underlined in html version) % \newcommand{\gs}[2]{\htmlref{#1}{#2}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % \e = EXAMPLE % % usage : % \h{continuum} % % output : % continuum (listed in the index, html label, list of examples) % %\def \e #1{{{\it #1}\label{#1}\index{#1}}} % %\def \e #1{{{\htmladdnormallink[#1]{\it #1}{../../s/a/table.xls}}\label{#1}\index{#1}}} % \def \e #1{{{\newtargetlink{\it #1}{../../s/a/table.xls}}\label{#1}\index{#1}}} % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % \es = EXAMPLES SPECIAL % % usage : % % \exs{Sierpi\'nski Tringle}{Sierpinski Triangle} % % output : % Sierpi\'nski Triangle, p. xxx (underlined in html version) % \newcommand{\es}[2]{\index{#2@#1}\label{#2}{\newtargetlink{\it #1}{../../s/a/table.xls}}} %\newcommand{\es}[2]{\index{#2@#1}% %\label{#2}{\htmladdnormallink[#2]{\it #1}{../../s/a/table.xls}} % % \newcommand{\exs}[2]{\hyperref[page]{ #2 }{ #1, p. }{ }{#2}} % \newcommand{\eS}[2]{\index{#2@#2}\label{#2}{\newtargetlink{\it #1}{../../s/a/table.xls}}} % \eS{(topologist's)}{topologist's sin surve} % % AND FOR THE LIST OF EXAMPLES : % \newcommand{\esLIST}[2]{\hyperref[page]{ #1 }{ #1, p. }{ }{#2} \newline} \newcommand{\eLIST}[2]{\hyperref[page]{ #2 }{ #2, p. }{ }{#2} \newline} \newcommand{\eSLIST}[2]{\hyperref[page]{ #2 }{ #2, p. }{ }{#2} \newline} % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % \h = HERE % % usage : % \h{continuum} % % output : % continuum (listed in the index, html label) % \def \h #1{{{\it #1}\label{#1}\index{#1}}} % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % \hs = HERE SPECIAL % % example % \hs {Sierpi\'nski Universal Plane Curve}{Sierpinski Universal Plane Curve} % tex output % {{{\it Sierpi\'nski Universal Plane Curve} % \label{Sierpinski Universal Plane Curve} % \index{Sierpinski Universal Plane Curve@Sierpi\'nski Universal Plane Curve}}} % % i.e. nicely printed, listed in index, labeled reference here % \newcommand{\hs}[2]{\index{#2@#1}% \label{#2}{\it #1}} % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % \hi = HERE INDEX ENTRY %\def \hi #1 #2{{{\it #1}\label{#1}\index{#2}}} % #1 text #2 index entry % example % \hi{end point} {point!end} % index output % point - end % \newcommand{\hi}[2]{\index{#2}% \label{#1}{\it #1}} % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % \hsi = HERE INDEX ENTRY %\def \hi #1 #2{{{\it #1}\label{#1}\index{#2}}} % #1 text #2 index entry % example % \hsi{end point} {end point}{point!end} % index output % point - end % % \hsi {Siepi\'nski curve}{Siepinski curve}{curve!Siepi\'nski} % \newcommand{\hsi}[3]{\index{#3}% \label{#2}{\it #1}} % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % cite -> city % % usage : % \city{sierpinski2001a}{Theorem 3.3, p.9} % % output % \cite[p.9]{sierpinski2001a}. % \newcommand{\city}[2]{\cite[#2]{#1}} % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % ITEMIZING !!!!! % % % \newcommand{\rostere}{\begin{enumerate}} \newcommand{\rosteree}{\end{enumerate}} \newcommand{\rosteri}{\begin{itemize}} \newcommand{\rosterii}{\end{itemize}} % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % % --------------------- % | Hyphenation Rules | % --------------------- % \hyphenation{con-tract-i-ble} \hyphenation{di-men-sion-al} \hyphenation{Hure-wicz} \hyphenation{Kras-in-kie-wicz} \hyphenation{ir-re-duc-i-ble} % % -------------------- % | Math Mode Macros | % -------------------- % \def\({\left(} % $ \( $ \def\){\right)} % $ \) $ \def\[{\left[} % $ \[ $ \def\]{\right]} % $ \] $ \newcommand{\mesh}[1]{mesh} \newcommand{\diam}[1]{diam} \def\lub{\operatorname{l.u.b}} % $ \lub $ \def\Cl{\overline} % set closure % $ \Cl{G_i} $ \def\ind{\operatorname{ind}} % $ \ind X $ % % --------------------------------- % | Accented-Foreign-Words Macros | % | (end with \ for white space) | % | ( e.g., \Cech\ cohomology ) | % --------------------------------- % % \def\Cech{\v{C}ech} % \def\Erdos{P.~Erd\H{o}s} % \def\Lindelof{Lindel\"of} % \def\Mackowiak{T.~Ma\'c\-ko\-wi\-ak} % \def\Sierpinski{Sierp\-i\'n\-ski} % % \newcommand{\Lim}{\mathrm{Lim}\,} \newcommand{\inter}{\mathrm{int}\,} \newcommand{\cl}{\mathrm{cl}\,} \newcommand{\bd}{\mathrm{bd}\,} \newcommand{\ord}{\mathrm{ord}\,} \newcommand{\card}{\mathrm{card}\,} \newcommand{\Cal}{\mathcal} %\newcommand{\diam}{\mathrm{diam}\,} \newcommand{\nbh}{neighborhood } \newcommand{\nbhs}{neighborhoods } \newcommand{\nbhsk}{neighborhoods. } \newcommand{\cm}{continuum } \newcommand{\cmb}{continuum} \newcommand{\cmk}{continuum. } \newcommand{\eps}{\varepsilon} \newcommand{\N}{\mathbb N} \newcommand{\R}{\mathbb R} \newcommand{\C}{\mathbb C} \newcommand{\I}{\mathbb I} % % % \renewcommand{\theequation}{\arabic{equation}} \renewcommand{\thefigure}{\arabic{chapter}.\arabic{section}.\arabic{subsection}} \addtocounter{chapter}{-1} % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% %%%%%%%%% START of CHAPTER %%%%%%%%% /c0000 %%%%%%%%% %%%%%%%%% \chapter{ PRELIMINARIES} All spaces considered here are assumed to be metric (if otherwise is not explicitly said) and all mappings are assumed to be continuous. Concepts and notions undefined in the text are used according to their meaning known from the literature. To aviod any doubt or misunderstanding, the reader can find the needed definition or explanation in Appendix A, where the items are alphabetically ordered. \par The following notation will be used. The symbol $\mathbb R$ stands for the space of real numbers, equipped with the natural topology. Thus $\mathbb R^2$ denotes the plane (usually supplied with a Cartesian coordinate system); equivalently, the plane can be denoted by $\mathbb C$ when considered as the set of all complex numbers. The closed unit interval $[0,1]$ of reals is denoted by $\mathbb I$, and $\mathbb N$ means the set of all positive integers. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% %%%%%%%%% START of CHAPTER %%%%%%%%% /c0001 %%%%%%%%% %%%%%%%%% \chapter{ LOCALLY CONNECTED CONTINUA} \section{ Basic examples: an arc, a circle, $n$-cube $\Bbb I^n$, Hilbert cube} \section{ Graphs} \begin{itemize} \item [(a)] Elementary examples: an arc, a circle, a noose, a figure eight, a $\theta$-curve \item [(b)] Acyclic graphs = trees. \item [(c)] planability of graphs; the Kuratowski graphs $K_{3,3}$, $K_5$; Whyburn's example of an open map from a planar onto a nonplanar graph \item [(d)] Problem of K\"onigsberg bridges; Euler's solution \end{itemize} \section{ Dendrites} A \g{dendrite} is a locally connected continuum containing no \g{simple closed curve}. By the \g{order of a point} $p$ in a dendrite $X$, writing $\ord(p, X)$\g{$\ord(p, X)$}, we mean the Menger-Urysohn order, see \cite[\S 51, I, p. 274]{Kuratowski1968a}, or equivalently, the \g{order in the classical sense}, i.e., the number of arcs emanating from $p$ and disjoint out of $p$ (see \cite[p. 229]{Charatonik1962a} and \cite[p. 301]{Lelek1961a}). \par In this chapter the term of an \g{end point} of a continuum is always used in the sense of a point of (Menger-Urysohn) order $1$, i.e., $p$ is an end point of $X$ provided that $\ord(p, X) = 1$. For dendrites this concept coincides with one of an \g{end point in the classical sense}, but not with the notion of an end point of an \g{arc-like} \g{continuum} as defined e.g. in \cite[p. 660]{Bing1951a}. \par Given a dendrite $X$, we denote by $E(X)$ the set of all end points of $X$ and by $R(X)$ the set of all its \gs{ramification points}{ramification point} (i.e., points of order at least 3). Various structural as well as mapping characterizations of dendrites are collected in \cite[Theorems 1.1 and 1.2, p. 228 and 230, respectively]{Charatonik+1998a}. See also \cite{Charatonik+2000a}. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% START %%%%%%%%% %%%%%%%%% /c0001/s0020/e0010 %%%%%%%%% %%%%%%%%% \subsection{ {Preliminary properties} } \setcounter{equation}{0} \label{dendrites-preliminary} The following properties of dendrites are known (see e.g. \cite[Chapter 10, p. 165]{Nadler1992a}, \cite [\S 51, VI, p. 300]{Kuratowski1968a} or \cite[p. 88]{Whyburn1942a}; compare also \cite[Theorems 1.3 and 1.5, p. 233]{Charatonik+1998a}). \rostere \item Every subcontinuum of a dendrite is a dendrite (thus each dendrite is hereditarily locally connected). \item Every connected subset of a dendrite is arcwise connected. \item In any dendrite $X$ the order of a point $p$ is equal to the number of components of $X \setminus \{p\}$. \item The set of all end points of a dendrite is $0$-dimensional. \item The set of all end points of a dendrite is a $G_\delta$-set. \item The set of all \gs{ordinary points}{ordinary point} (i.e., points of order 2) of a dendrite $X$ is dense in $X$. \item Every dendrite has at most countably many ramification points. \item If $m$ is the order of a ramification point in a dendrite, then $m \in \{3,4,\dots,\omega\}$. \item No dendrite contains points of order $\aleph_0$ or $\mathfrak c$ (i.e., each dendrite is a \g{regular continuum}). \par \item In any dendrite the set of all its end points is dense if and only if the set of all its ramification points is dense. \item Dendrites are \gs{absolute retracts}{absolute retract}. In fact, the class of all 1-dimensional absolute retracts coincide with the class of dendrites, \cite[Corollary 13.5, p. 138]{Borsuk1967a}. \item A continuum $X$ is a dendrite if and only if for every compact space (continuum) $Y$ and for every \gs{light}{light mapping} \g{confluent mapping} $f: Y \to f(Y)$ such that $X \subset f(Y)$ there is a copy $X'$ of $X$ in $Y$ for which the restriction $f|X': X' \to X$ is a homeomorphism, \cite{Charatonik+XXXXb}. \rosteree \STORY{../s/c0001/s0020/e0010} %%%%%%%%% %%%%%%%%% END %%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% START %%%%%%%%% %%%%%%%%% /c0001/s0020/e0020 %%%%%%%%% %%%%%%%%% \subsection{ {Simple examples} } \setcounter{equation}{0} The simplest examples of dendrites are the \g{arc} and \gs{$n$-ods}{n-od} for $n \in \{3, 4, \dots \}$. All they are trees. By a \g{tree} we mean a dendrite with finitely many end points, or equivalently, a dendrite containing no points of order $\omega$ and having the set of all ramification points finite. \STORY{../s/c0001/s0020/e0020} %%%%%%%%% %%%%%%%%% END %%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% START %%%%%%%%% %%%%%%%%% /c0001/s0020/e0030 %%%%%%%%% %%%%%%%%% \subsection{ {The locally connected fan} } \setcounter{equation}{0} \label{ex-Fomega} Another simple example is the \e{locally connected fan} $F_\omega$. It is defined as the union of countably many straight line segments of length tending to $0$, emanating from a point $p$ and disjoint out of $p$. Thus $F_\omega$ is the dendrite with only one ramification point whose order is $\omega$. \FIGURE[scale=1]{}{../s/c0001/s0020/e0030/}{A}{locally connected fan $F_\omega$} %Figure as in \cite[Fig. 1, p. 9]{Charatonik+1990a} The following mapping properties of $F_\omega$ are known. \rostere \item\label{ex-Fomega-univ} $F_\omega$ is \gs{universal}{universal space} in the class of all $n$-ods for $n \in \mathbb N$. \item Each \gs{open image}{open mapping} of $F_\omega$ is homeomorphic to $F_\omega$, \cite[Proposition 9.4, p. 42]{Charatonik+1990a}. \item A \g{confluent mapping} defined on $F_\omega$ is open if and only if it is \gs{light}{light mapping}, \cite[Theorem 9.6, p. 42]{Charatonik+1990a}. \rosteree \STORY{../s/c0001/s0020/e0030} %%%%%%%%% %%%%%%%%% END %%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% START %%%%%%%%% %%%%%%%%% /c0001/s0020/e0040 %%%%%%%%% %%%%%%%%% \subsection{ {The locally connected combs} } \setcounter{equation}{0} \label{ex-lccombs} The next two examples are defined in \cite[p. 3]{Arevalo+2001a}. Let $\overline {pq}$ stand for the straight line segment from $p$ to $q$ in the plane. In the plane, let $a = (0,0), a_n = (1/n, 1/n), b_n = (1/n, 0)$ for each $n \in \mathbb N$, and $c = (-1,0)$. Define $$W_R = \overline{ab_1} \cup \bigcup \{\overline{a_nb_n}: n \in \mathbb N\} \quad \text{and} \quad W = \overline{ca} \cup W_R.$$ The dendrites $W_R$ and $W$ are called the \es{comb $W_R$}{comb WR} and the \es{comb $W$}{comb W} , or (generally) \es{locally connected combs}{locally connected comb} . See Figures A--B. \FIGURE[scale=1]{}{../s/c0001/s0020/e0040/}{A}{comb $W_R$} \FIGURE[scale=1]{}{../s/c0001/s0020/e0040/}{B}{comb $W$} %Figure as in \cite[Fig. 1, p. 2]{Arevalo+2001a} (for $W$). The dendrites $F_\omega$, $W_R$ and $W$ are exploited in the following characterizations. \rostere \item A dendrite is a tree if and only if it contains neither a copy of $F_\omega$ nor of $W_R$, \cite[Theorem 3.1, p. 3]{Arevalo+2001a}. \item\label{ex-lccombs-clends} A dendrite has the set of all its end points closed if and only if it contains neither a copy of $F_\omega$ nor of $W$, \cite[Corollary 5.4, p. 11]{Arevalo+2001a}. \rosteree \STORY{../s/c0001/s0020/e0040} %%%%%%%%% %%%%%%%%% END %%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% START %%%%%%%%% %%%%%%%%% /c0001/s0020/e0050 %%%%%%%%% %%%%%%%%% \subsection{ {Universal dendrites} } \setcounter{equation}{0} As it was mentioned in Property~\ref{ex-Fomega-univ} in~\ref{ex-Fomega} the \g{locally connected fan} is universal in the class of all $n$-ods, where $n \in \mathbb N$. Now we recall the construction of the \gs{Wa\.zewski universal dendrite}{Wazewski universal dendrite} $D_\omega$ and some other dendrites that are universal in various classes of these curves. \STORY{../s/c0001/s0020/e0050} %%%%%%%%% %%%%%%%%% END %%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% START %%%%%%%%% %%%%%%%%% /c0001/s0020/e0060 %%%%%%%%% %%%%%%%%% \subsection{ {Wa\.zewski universal dendrite} } \setcounter{equation}{0} By the \es{Wa\.zewski universal dendrite}{Wazewski universal dendrite} $D_\omega$ we mean a dendrite $D_\omega$ such that each ramification point of $D_\omega$ is of order $\omega$ and for each arc $A \subset D_\omega$ the set of ramification points of $D_\omega$ which belong to $A$ is dense in $A$. Its construction, known from \cite[Chapter K, p. 137]{Whyburn1942a} (compare also \cite[Chapter X, Section 6, p. 318]{Menger1932a}), is the following. Let $X^{(\omega)}_1 = F_\omega$. At the midpoint $c$ of each maximal \g{free arc} contained in $X^{(\omega)}_1$ (obviously the arc is a straight line segment) attach a sufficiently small copy of $F_\omega$ so that $c$ is the only common point of $X^{(\omega)}_1$ and of the attached copy. Denote by $X^{(\omega)}_2$ the union of $X^{(\omega)}_1$ and of all attached copies. Thus $X^{(\omega)}_2$ is a dendrite. At the midpoint of each maximal free arc contained in $X^{(\omega)}_2$ we perform the same construction, i.e., we attach a sufficiently small copy of $F_\omega$ so that $m$ is the only common point of $X^{(\omega)}_2$ and of the attached copy. Denote by $X^{(\omega)}_3$ the union of $X^{(\omega)}_2$ and of all attached copies. Thus $X^{(\omega)}_3$ is a dendrite. Continuing in this way we obtain an increasing sequence of dendrites $X^{(\omega)}_1 \subset X^{(\omega)}_2 \subset \dots \subset X^{(\omega)}_n \subset X^{(\omega)}_{n+1} \subset \dots $. The construction can be done in the plane in such a way that the limit continuum $D_\omega$ defined by $$D_\omega = \cl(\bigcup \{X^{(\omega)}_n: n \in \mathbb N\})$$ is again a dendrite. See Figure A. \FIGURE[scale=1]{}{../s/c0001/s0020/e0060/}{A}{Wa\.zewski universal dendrite $D_\omega$} %Figure 13.3.1 here For another construction of $D_\omega$ (using inverse limits) see \cite[10.37, p. 181-185]{Nadler1992a}. The following properties of $D_\omega$ are known. \rostere \item $D_\omega$ is \gs{universal}{universal space} in the class of all dendrites (see e.g. \cite[10.37, p. 181-185]{Nadler1992a}). \item $D_\omega$ is embeddable in the plane (in fact, it is constructed in the plane). \item Each \gs{open image}{open mapping} of $D_\omega$ is homeomorphic to $D_\omega$ (see \cite[Theorem 1, p. 490]{Charatonik1980a}). \item $D_\omega$ is \gs{homogeneous with respect to monotone mappings} {homogeneous with respect to a class of mappings}, \cite[Theorem 7.1, p. 186]{Charatonik1991a}. \rosteree For other mapping properties of $D_\omega$, in particular ones related to the action of the group of autohomeomorphisms on $D_\omega$, see \cite{Charatonik1995a}. \STORY{../s/c0001/s0020/e0060} %%%%%%%%% %%%%%%%%% END %%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% START %%%%%%%%% %%%%%%%%% /c0001/s0020/e0070 %%%%%%%%% %%%%%%%%% \subsection{ {Universal dendrites of order $m$} } \setcounter{equation}{0} \label{ex-universal-dendrite-m} Let $m \in \{3, 4, \dots, \omega\}$. By the \es{standard universal dendrite of order $m$}{standard universal dendrite of order m} we mean a \es{dendrite $D_m$}{dendrite Dm} such that each ramification point of $D_m$ is of order $m$ and for each arc $A \subset D_m$ the set of ramification points of $D_m$ which belong to $A$ is dense in $A$. Their constructions, known again from \cite[Chapter K, p. 137]{Wazewski1923a} (see also \cite[(4), p. 168]{Charatonik1991a}; for the inverse limit construction see \cite[p. 491]{Charatonik1980a}), mimic that of the Wa\.zewski universal dendrite $D_\omega$, but instead of copies of $F_\omega$ we use copies of $m$-ods at each step of the construction. %More precisely, we take $X^{(m)}_1$ as an $m$-od (being the union of %straight line segments). Having defined $X^{(m)}_n$ for some $n \in %\mathbb N$ (that is the union of straight line segments in the plane) we %attach at the midpoint of each maximal free arc contained in $X^{(m)}_n$ a %sufficiently small copy of $(m - 2)$-od so that $c$ is the only common %point of $X^{(\omega)}_1$ and of the attached copy. Then $X^{(m)}_{n+1}$ is %defined as the union of $X^{(m)}_n$ and of all the attached copies of $(m - %2)$-ods. Note that each ramification point of $X^{(m)}_{n+1}$ is of order %$m$ by construction. All this can be done in the plane in such a way that %the limit continuum $D_\omega$ defined by %$$D_m = \mathrm{cl}\,(\bigcup \{X^{(m)}_n: n \in \mathbb N\})$$ %is again a dendrite. In particular, if $m = \omega$, then the standard %universal dendrite of order $\omega$ is just the Wa\.zewski dendrite %$D_\omega$. See Figures A--C for the standard universal dendrites $D_3$, $D_4$ and $D_6$. \FIGURE[scale=1]{}{../s/c0001/s0020/e0070/}{A}{standard universal dendrite $D_3$} \GIF[scale=1]{}{../s/c0001/s0020/e0070/}{AA}{standard universal dendrite $D_3$ - an animation} \FIGURE[scale=1]{}{../s/c0001/s0020/e0070/}{B}{standard universal dendrite $D_4$} \GIF[scale=1]{}{../s/c0001/s0020/e0070/}{BB}{standard universal dendrite $D_4$ - an animation} \GIF[scale=1]{}{../s/c0001/s0020/e0070/}{BBB}{standard universal dendrite $D_4$ produced as an intersection - an animation} \FIGURE[scale=1]{}{../s/c0001/s0020/e0070/}{C}{standard universal dendrite $D_6$} The standard universal dendrites $D_m$ have the following properties. \rostere \item For each $m \in \{3, 4, \dots, \omega\}$ $D_m$ is \gs{universal}{universal space} in the class of all dendrites for which the order of their ramification points is less than or equal to $m$, see e.g. \cite[Chapter 10, \S 6, p. 322]{Menger1932a}. \item If $m, n \in \mathbb N$ with $3 \le m < n$, then there exists an \g{open mapping} of $D_n$ onto $D_m$, \cite[Theorem 2, p. 492]{Charatonik1980a}. \item Among all standard universal dendrites $D_m$ only $D_3$ and $D_\omega$ are homeomorphic with all their \gs{open images}{open mapping}, \cite[Corollary, p. 493]{Charatonik1980a}. \item For each $m \in \{3, 4, \dots, \omega\}$ a \gs{monotone surjection} {monotone mapping} of $D_m$ onto itself is a \g{near homeomorphism} if and only if $m = 3$, \cite[Corollary 5.5, p. 178]{Charatonik1991a}. \item\label{ex-ex-universal-dendrite-m-twomonoequi} Any two standard universal dendrites $D_m$ and $D_n$ of some orders $m, n \in \{3, 4, \dots, \omega\}$ are \gs{monotonely equivalent}{monotonely equivalent spaces}, \cite[Corollary 6.6, p. 180]{Charatonik1991a}. \item\label{ex-universal-dendrite-m-dmmonohom} For each $m \in \{3, 4, \dots, \omega\}$ the dendrite $D_m$ is \gs{monotonely homogeneous}{homogeneous with respect to monotone mappings} \cite[Theorem 7.1, p. 186]{Charatonik1991a}. \rosteree Other mapping properties of the standard universal dendrites $D_m$ can be found e.g. in \cite{Charatonik1980a}, \cite{Charatonik1991a}, \cite {Charatonik1995a}, \cite {Charatonik+1997a}, \cite{Charatonik+1998a}, \cite{Charatonik+1994a} and \cite{CharatonikW+1994a}. \STORY{../s/c0001/s0020/e0070} %%%%%%%%% %%%%%%%%% END %%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% START %%%%%%%%% %%%%%%%%% /c0001/s0020/e0080 %%%%%%%%% %%%%%%%%% \subsection{ {Other universal dendrites} } \setcounter{equation}{0} \label{ex-other-universal-dendrites} For a given nonempty set $S \subset \{3, 4, \dots, \omega\}$ we denote by $D_S$ any dendrite $X$ satisfying the following two conditions: \rosteri \item[(a)] if $p$ is a ramification point of $X$, then $\ord(p,X) \in S$; \item[(b)] for each arc $A$ contained in $X$ and for every $m \in S$ there is in $A$ a point $p$ with $\ord(p,X) = m$. \rosterii It is shown in \cite[Theorem 6.2, p. 229]{CharatonikW+1994a} that the \es{dendrite $D_S$}{dendrite DS} is topologically unique, i.e., if two dendrites satisfy conditions (a) and (b) with the same set $S \subset \{3, 4, \dots, \omega\}$, then they are homeomorphic. The dendrite $D_S$ is called the \es{standard universal dendrite of orders in $S$}{standard universal dendrite of orders in S} . If $S$ is a singleton $\{m\}$, then $D_S$ is just the \gs{standard universal dendrite $D_m$}{standard universal dendrite Dm} defined previously. The following properties of dendrites $D_S$ are known (see \cite[Theorems 6.4 and 6.6-6.8, p. 230; Corollary 6.10, p. 232]{CharatonikW+1994a}). \rostere \item\label{ex-other-universal-dendrites-spsh} For any nonempty subset $S \subset \{3, 4, \dots, \omega\}$, the dendrite $D_S$ is \g{strongly pointwise self-homeomorphic}. \item If $\omega \in S$, then the dendrite $D_S$ is \gs{universal}{universal space} for the family of all dendrites. \item If the set $S$ is finite with $\max S = m$, then $D_S$ is universal for the family of all dendrites having orders of ramification points at most $m$. \item\label{ex-other-universal-dendrites-univfinord} If the set $S$ is infinite and $\omega \notin S$, then $D_S$ is universal for the family of all dendrites having finite orders of ramification points. \item Nonconstant \gs{open images}{open mapping} of standard universal dendrites $D_S$ are homeomorphic to $D_S$ if and only if $S$ is a nonempty subset of $\{3, \omega\}$. \item For any nonempty subset $S \subset \{3, 4, \dots, \omega\}$ and for an arbitrary dendrite $Y$ there exists a \g{monotone mapping} from $D_S$ onto $Y$, \cite[Theorem 2.22, p. 239]{Charatonik+1998a}. \item\label{ex-other-universal-dendrites-DSmonohom} For any nonempty subset $S \subset \{3, 4, \dots, \omega\}$ the dendrite $D_S$ is \gs{monotonely homogeneous}{homogeneous with respect to monotone mappings}, \cite[Theorem 3.3, p. 292]{Charatonik1996a}. \rosteree \STORY{../s/c0001/s0020/e0080} %%%%%%%%% %%%%%%%%% END %%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% START %%%%%%%%% %%%%%%%%% /c0001/s0020/e0090 %%%%%%%%% %%%%%%%%% \subsection{ {The dendrite $P$} } \setcounter{equation}{0} \label{ex-dendrite-P} In connection with the characterization of dendrites having closed set of their end points (see Property~\ref{ex-lccombs-clends} in~\ref{ex-lccombs} above) the classes of those dendrites that do not contain a copy of $F_\omega$ and those that do not contain a copy of the \gs{comb $W$}{comb W} are of some interest (see \cite[p. 8]{Arevalo+2001a}). The first one is just the class of dendrites with finite orders of ramification points, and it is known to have a universal element according to Property~\ref{ex-other-universal-dendrites-univfinord} in~\ref{ex-other-universal-dendrites}. The other class has also a universal element $P$. Its construction, given in \cite[p. 9]{Arevalo+2001a} is the following. Let $p$ be any point in the plane $\mathbb R^2$ and let $p_1, p_2, \dots$ be a sequence of points in $\mathbb R^2$ tending to $p$ and such that no three of the points $p, p_1, p_2, \dots$ are collinear. Define $P_1 = \bigcup \{\overline{pp_n}: n \in \mathbb N\}$. Then $P_1$ is homeomorphic to $F_\omega$. For every $n, m \in \mathbb N$, let $p_{nm}$ be a point in $\mathbb R^2 \setminus P_1$ such that for any fixed $n$ the sequence $p_{n1}, p_{n2}, \dots$ tends to $p_n$, no three of the points $p_n, p_{n1}, p_{n2}, \dots$ are collinear, and that the continuum $P_2 = P_1 \cup \bigcup \{\overline{p_np_{nm}}: n, m \in \mathbb N\}$ is a dendrite. We continue constructing dendrites $P_3, P_4, \dots$ in the same manner, such that we get an increasing sequence of dendrites $P_1 \subset P_2 \subset P_3 \subset \dots \subset P_n \subset P_{n+1} \subset \dots$ in $\mathbb R^2$ having the property that the closure of their union is also a dendrite. Finally define $$ P = \cl (\bigcup\{P_n: n \in \mathbb N\}). $$ The following properties of the \es{dendrite $P$}{dendrite P} are proved in \cite[Theorems 4.5 and 4.6, p. 10]{Arevalo+2001a}. \rostere \item A dendrite $X$ is homeomorphic to the dendrite $P$ if and only if each of the following conditions is satisfied: \rostere \item $X$ contains no copy of \gs{$W$}{comb W}; \item the set of all end points of $X$ is contained in the closure of the set of all ramification points of $X$; \item all ramification points of $X$ are of order $\omega$. \rosteree \item The dendrite $P$ is \gs{universal}{universal dendrite} in the class of all dendrites containing no copy of $W$. \rosteree See Figure A. \FIGURE[scale=1]{}{../s/c0001/s0020/e0090/}{A}{dendrite $P$} \GIF[scale=1]{}{../s/c0001/s0020/e0090/}{AA}{dendrite $P$ - an animation} \STORY{../s/c0001/s0020/e0090} %%%%%%%%% %%%%%%%%% END %%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% START %%%%%%%%% %%%%%%%%% /c0001/s0020/e0100 %%%%%%%%% %%%%%%%%% \subsection{ {Self-homeomorphic dendrites} } \setcounter{equation}{0} Below we present some examples of dendrites that illustrate relations between variants of self-homeomorphic properties. \rostere \item There is a \g{strongly pointwise self-homeomorphic} dendrite which is not of the form \gs{$D_S$}{dendrite DS} for some $S \subset \{3, 4, \dots, \omega\}$, see \cite[Example 6.11, p. 232, and Fig. 4, p. 233]{CharatonikW+1994a}. Note that, according to Property~\ref{ex-other-universal-dendrites-spsh} in~\ref{ex-other-universal-dendrites} each dendrite $D_S$ is strongly pointwise self-homeomorphic. This example shows that the opposite implication does not hold. Some additional properties of the example are shown in \cite[Example 2.1, p. 152]{Pyrih1999b}. See Figure A. \FIGURE[scale=1]{}{../s/c0001/s0020/e0100/}{A}{an example of self-homeomorphic dendrite} \item There is a \g{strongly self-homeomorphic} and \g{pointwise self-homeomorphic} dendrite which is not \g{strongly pointwise self-homeomorphic}, see \cite[Example 6.12, p. 233]{CharatonikW+1994a}. \item There is a strongly self-homeomorphic dendrite which is not pointwise self-homeomorphic, see \cite[Example 2.1, p. 572]{Pyrih1999a}. Dendrites having this property are studied in \cite{Charatonik+XXXXc}. See Figure B. \FIGURE[scale=1]{}{../s/c0001/s0020/e0100/}{B}{an example of strongly self-homeomorphic dendrite} \rosteree \STORY{../s/c0001/s0020/e0100} %%%%%%%%% %%%%%%%%% END %%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% START %%%%%%%%% %%%%%%%%% /c0001/s0020/e0110 %%%%%%%%% %%%%%%%%% \subsection{ {Monotone equivalence and monotone homogeneity} } \setcounter{equation}{0} As it was mentioned previously, any two universal dendrites $D_m$ and $D_n$ are \g{monotonely equivalent} (see Property~\ref{ex-ex-universal-dendrite-m-twomonoequi} in~\ref{ex-universal-dendrite-m}). However, the family of all dendrites which are monotonely equivalent to a dendrite $D_m$ contains also other members. To characterize the family we have to construct a special dendrite, denoted $L_0$. \STORY{../s/c0001/s0020/e0110} %%%%%%%%% %%%%%%%%% END %%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% START %%%%%%%%% %%%%%%%%% /c0001/s0020/e0120 %%%%%%%%% %%%%%%%%% \subsection{ {Omiljanowski dendrite} } \setcounter{equation}{0} The \e{Omiljanowski dendrite} has been constructed by K. Omiljanowski in \cite[Example 6.9, p. 182]{Charatonik1991a} (see also \cite[Remark 6.11 and Theorem 6.12, p. 183]{Charatonik1991a}). It is defined as the closure of the union of an increasing sequence of dendrites in the plane. We start with the unit straight line segment denoted by $L_1$. Divide $L_1$ into three equal subsegments and in the middle of them, $M$, locate a thrice diminished copy of the Cantor ternary set $C$. At the midpoint of each contiguous interval $K$ to $C$ (i.e., a component $K$ of $M \setminus C$) we erect perpendicularly to $L_1$ a straight line segment whose length equals length of $K$. Denote by $L_2$ the union of $L_1$ and of all erected segments (there are countably many of them, and their lengths tend to zero). We perform the same construction on each of the added segments: divide such a segment into three equal parts, locate in the middle part $M$ a copy of the Cantor set $C$ properly diminished, at the midpoint of any component $K$ of $M \setminus C$ construct a perpendicular to $K$ segment as long as $K$ is, and denote by $L_3$ the union of $L_2$ and of all attached segments. Continuing in this manner we get a sequence of dendrites $L_1 \subset L_2 \subset L_3 \subset \dots $. Finally we put $$ L_0 = \mathrm{cl}\, (\bigcup \{L_i: i \in \Bbb N\}) $$ and call it a \es{dendrite $L_0$}{dendrite L0} . See Figure A. \FIGURE[scale=1]{}{../s/c0001/s0020/e0120/}{A}{dendrite $L_0$} % PP: please, produce a picture, if possible. The Omiljanowski dendrite $L_0$ has the following properties, \cite[Example 6.9, p. 182]{Charatonik1991a}. \rostere \item All ramification points of $L_0$ are of order 3. \item The set $R(L_0)$ of all ramification points of $L_0$ is discrete (thus nowhere dense). \item The set of all end points of $L_0$ is nowhere dense. \item For each maximal arc $A$ in $L_0$ the closure of the set $A \cap R(L_0)$ contains a homeomorphic copy of the Cantor set. \item $L_0$ is \g{monotonely equivalent} to the \gs{dendrite $D_3$}{dendrite D3}. \item The following conditions are equivalent for a dendrite $X$ (see \cite[Theorem 6.14, p. 185]{Charatonik1991a} and \cite[Theorem 5.35, p. 17]{Charatonik+1994a}; compare \cite[Theorem 2.20, p. 238]{Charatonik+1998a}): \rostere \item $X$ is monotonely equivalent to $D_3$; \item $X$ is monotonely equivalent to \gs{$D_\omega$}{dendrite Domega}; \item $X$ is monotonely equivalent to \gs{$D_m$}{dendrite Dm} for each $m \in \{3, 4, \dots, \omega\}$; \item $X$ is monotonely equivalent to \gs{$D_S$}{dendrite DS} for each nonempty set $S \subset \{3, 4, \dots, \omega\}$; \item $X$ is monotonely equivalent to every dendrite $Y$ having dense set $R(Y)$ of its ramification points; \item $X$ is monotonely equivalent to some dendrite $Y$ having dense set $R(Y)$ of its ramification points; \item $X$ contains a homeomorphic copy of every dendrite $L$ such that its set $R(L)$ of ramification points is discrete and consists of points of order $3$ exclusively; \item $X$ contains a homeomorphic copy of the dendrite $L_0$. \rosteree According to Property~\ref{ex-other-universal-dendrites-DSmonohom} in~\ref{ex-other-universal-dendrites} (see also Property~\ref{ex-universal-dendrite-m-dmmonohom} in~\ref{ex-universal-dendrite-m}) any dendrite $D_S$ is \gs{monotonely homogeneous}{homogeneous with respect to monotone mappings}. Another, less restrictive, sufficient condition for monotone homogeneity of a dendrite is the following (see \cite[Proposition 15, p. 364]{Charatonik+1997a}). \item If for a dendrite $X$ the set $R(X)$ of its ramification points is dense in $X$, then $X$ is \gs{monotonely homogeneous} {homogeneous with respect to monotone mappings}. The condition $\cl R(X) = X$, being sufficient, is far from being necessary. Namely the Omiljanowski dendrite $L_0$ has the set $R(L_0)$ discrete, and it is \gs{monotonely homogeneous} {homogeneous with respect to monotone mappings}. Moreover, the following statement holds, \cite[Proposition 20, p. 366]{Charatonik+1997a}. \item If a dendrite $X$ contains a homeomorphic copy of $L_0$, then $X$ is \gs{monotonely homogeneous}{homogeneous with respect to monotone mappings}. \rosteree It would be interesting to know if the converse to the above statement holds true, i.e., if containing the dendrite $L_0$ characterizes monotonely homogeneous dendrites. In other words, we have the following question. \textbf{Question.} Does every \gs{monotonely homogeneous} {homogeneous with respect to monotone mappings} dendrite contain a homeomorphic copy of $L_0$? The above question is closely related to a more general problem. \textbf{Problem.} Give any structural characterization of \gs{monotonely homogeneous} {homogeneous with respect to monotone mappings}. \STORY{../s/c0001/s0020/e0120} %%%%%%%%% %%%%%%%%% END %%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% START %%%%%%%%% %%%%%%%%% /c0001/s0020/e0130 %%%%%%%%% %%%%%%%%% \subsection{ {Chaotic and rigid dendrites} } \setcounter{equation}{0} We start presentation of examples related to chaoticity with an example of a dendrite which, while not chaotic, is closely related to the property, and which has been considered as a pattern to construct other examples having various related properties. \STORY{../s/c0001/s0020/e0130} %%%%%%%%% %%%%%%%%% END %%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% START %%%%%%%%% %%%%%%%%% /c0001/s0020/e0140 %%%%%%%%% %%%%%%%%% \subsection{ {Miller dendrite} } \setcounter{equation}{0} Recall that a space is said to be \gs{ incompressible}{incompressible space} if it is homeomorphic to no proper subspace of itself. The circle $S^1$ is an example of such a space. In 1925 Zarankiewicz \cite{Zarankiewicz1925a} asked whether there exists an incompressible dendrite. The question has been answered in the positive in 1932 by E. W. Miller \cite{Miller1932a}. The reader is referred to the original Miller's paper for the (rather long) construction of the example. The main idea of the proof is based on the following implication, \cite[Theorem, p. 831]{Miller1932a}. If a dendrite $X$ contains a subset $K$ such that \rosteri \item [A.] each point of $K$ is fixed with respect to any homeomorphism of $X$ onto a subcontinuum of $X$; \item [B.] each point of $X$ which is not an end point of $X$ lies in an arc contained in $X$ and having its end points in $K$, \rosterii then $X$ is incompressible. The \es{Miller dendrite $S$}{Miller dendrite S} has the following properties, \cite{Miller1932a}. \rostere \item $S$ contains a subset $K$ having properties A and B above, so it is incompressible. \item $S$ is not \gs{chaotic}{chaotic space}. \item $S$ contains open arcs as open subsets, hence it is not \gs{rigid}{rigid space}. \rosteree \STORY{../s/c0001/s0020/e0140} %%%%%%%%% %%%%%%%%% END %%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% START %%%%%%%%% %%%%%%%%% /c0001/s0020/e0150 %%%%%%%%% %%%%%%%%% \subsection{ {Dendrites of de Groot-Wille type} } \setcounter{equation}{0} An example of a rigid dendrite has been roughly described in \cite{Groot+1958a}. Its construction is recalled below (see also \cite[Chapter III, Section 5 , p. 227]{Charatonik1979a}). For any $i \in \mathbb N$ let $T_i$ denote the $i$-od being the union of $i$ straight line unit segments emanating from a point called the \g{origin} of $T_i$. We proceed by induction. Define $Y_0$ as the unit segment. Let $x$ be the midpoint of $Y_0$, and define $Y_1$ as the union of $Y_0$ and a diminished copy of $T_{j_1} = T_1$ so that the diameter of this copy is less than $1/2$ and that $Y_0 \cap T_1 = \{x\}$. Assume that a tree $Y_n$ is defined such that it is the union of finitely many straight line segments and contains (properly diminished) copies of $T_1, T_2, \dots, T_{j_n}$, i.e., of the first $j_n$ terms of the sequence $\{T_i\}$. To define $Y_{n+1}$ consider all maximal free segments in $Y_n$. Note that there are finitely many, say $m(n)$ of them. Let $x$ denote the midpoint of any of these segments. With each point $x$ so defined we associate, in a one-to-one way, a set $T_i$ taken from the $m(n)$ consecutive terms of the sequence $\{T_i\}$, i.e., we use in this step of the construction the sets $T_{j_n + 1}, T_{j_n + 2}, \dots, T_{j_{n + 1}}$, where $j_{n+1} = j_n + m(n)$. We take each midpoint as the origin of a properly diminished copy of $T_i$, where $i \in \{j_n + 1, j_n + 2, \dots, j_{n + 1}\}$ in such a way that the diameter of the copy of $T_i$ is less than $1/2^{n+1}$ and that $Y_n$ has only the point $x$ in common with the attached copy of $T_i$. All this can clearly be done so carefully that the resulting set $Y_{n+1}$ is a tree and the limit continuum $$ Y = \cl(\bigcup \{Y_n: n \in \mathbb N\})$$ is a dendrite. See Figure A. \FIGURE[scale=1]{}{../s/c0001/s0020/e0150/}{A}{$Y_3$} \GIF[scale=1]{}{../s/c0001/s0020/e0150/}{AA}{$Y_3$} The following properties of the dendrite $Y$ are shown in \cite[Statement 10, p. 229]{Charatonik1979a}. \rostere \item\label{ex-Groot-Edens} The set of all end points of $Y$ is dense in $Y$. \item For each $n \in \mathbb N$ the set of all points of $Y$ that are of order at least $n$ is dense in $Y$. \item\label{ex-Groot-only1} For each natural $n \ge 3$ the dendrite $Y$ contains exactly one point of order $n$. \item For each point $p \in Y$ and for each component $C$ of $Y \setminus \{p\}$ the continuum $C \cup \{p\}$ contains a homeomorphic copy of $Y$. \item\label{ex-Groot-notsr} $Y$ is not \g{strongly rigid}. \item\label{ex-Groot-chao} $Y$ is \gs{chaotic}{chaotic space}. \rosteree Recall that for any dendrite conditions~\ref{ex-Groot-Edens} and~\ref{ex-Groot-only1} imply~\ref{ex-Groot-notsr} and~\ref{ex-Groot-chao}, see \cite[Theorem 13, p. 22]{Charatonik1999a}. However, neither the above mentioned conditions, nor a rough description given in \cite{Groot+1958a}, nor the one presented in \cite{Charatonik1979a} lead to a uniquely determined dendrite, because the constructed dendrite $Y$ depends on a function that assigns the consecutive $i$-ods $T_i$ (used in the succesive steps of the construction) to the midpoints of the maximal free arcs in the trees $Y_n$. Thus we refer to any of the dendrites $Y$ obtained in this way as to a \e{dendrite of de Groot-Wille type} . An example of a dendrite of de Groot-Wille type that is \gs{chaotic}{chaotic space} but not \g{openly chaotic} is constructed in \cite[Theorem 3.10, p. 646]{Charatonik2000a}. \STORY{../s/c0001/s0020/e0150} %%%%%%%%% %%%%%%%%% END %%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% START %%%%%%%%% %%%%%%%%% /c0001/s0020/e0160 %%%%%%%%% %%%%%%%%% \subsection{ {Modified Miller dendrites} } \setcounter{equation}{0} The methods used by Miller to construct his example $S$ has also been applied to get a modification $D$ of $S$ in \cite[Chapter IV, p. 230]{Charatonik1979a} (an outline of this construction is in \cite{Charatonik1977a}). Below we summarize properties of this example, and recall an extension of this construction. \rostere \item There exists a dendrite $D$ such that: \rostere \item each point of $D$ is of order nor greater than $4$; \item for each $n \in \{1, 2, 3, 4\}$ the set of all points of $D$ which are of order $n$ is dense in $D$; \item $D$ is \gs{strongly rigid}{strongly rigid space}; \item $D$ is \gs{chaotic}{chaotic space}. \rosteree \item For any two integers $m$ and $n$ with $3 \le m < n$ there exists a dendrite $X(m,n)$ such that (see \cite[Theorem 5.5, p. 185]{Charatonik+1996a} and compare also \cite[Theorem 27, p. 24]{Charatonik1999a}): \rostere \item $\ord(x, X(m,n)) \in \{1,2,m, n\}$ for each $x \in X(m,n)$. \item Every arc in $X(m,n)$ contains a point of order $m$ in $X(m,n)$. \item Each of the sets: of all end points of $X(m,n)$, of all points of order $m$ in $X(m,n)$ and of all points of order $n$ in $X(m,n)$ is dense in $X(m,n)$. \item $X(m,n)$ is \gs{strongly chaotic}{strongly chaotic space}. \rosteree Dendrite $D$ is called a \e{modified Miller dendrite} . (For a further modification of the above construction that leads to chaotic, strongly rigid, openly rigid and not strongly chaotic dendrites, see \cite[Theorem 3.14, p. 650]{Charatonik2000a}.) Finally recall dendrites constructed in \cite[Examples 33 and 35, p. 28]{Charatonik1999a}. \item For each natural $n \ge 5$, there exists a strongly rigid and not chaotic dendrite, all points of which are of order at most $n$ (\cite[Example 33, p. 28]{Charatonik1999a}). \item There exists a rigid dendrite which is neither chaotic nor strongly rigid (\cite[Example 35, p. 28]{Charatonik1999a}). \rosteree \STORY{../s/c0001/s0020/e0160} %%%%%%%%% %%%%%%%%% END %%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% START %%%%%%%%% %%%%%%%%% /c0001/s0020/e0170 %%%%%%%%% %%%%%%%%% \subsection{ {Dendrites with the closed set of end points} } \setcounter{equation}{0} Dendrites in this class have been already characterized (see Property~\ref{ex-lccombs-clends} in~\ref{ex-lccombs}). Besides any tree, the locally connected comb $W_R$ \g{comb $W_R$} is an example of a dendrite having the set of its end points closed (and countable). Below we recall a member of the considered class having uncountably many end points. \STORY{../s/c0001/s0020/e0170} %%%%%%%%% %%%%%%%%% END %%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% START %%%%%%%%% %%%%%%%%% /c0001/s0020/e0180 %%%%%%%%% %%%%%%%%% \subsection{ {Gehman dendrite} } \setcounter{equation}{0} One of the classical examples of dendrites with its set of end points closed is the \e{Gehman dendrite} . It can be described as a dendrite $G$ having the set $E(G)$ homeomorphic to the Cantor ternary set $C$ in $[0,1]$ such that all ramification points of $G$ are of order 3 (see \cite[the example on p. 42]{Gehman1925a}; see also \cite[p. 422-423]{Nikiel1983a} for a geometrical description; compare \cite[p. 82]{Nikiel1989a} and \cite[Example 10.39, p. 186]{Nadler1992a}). See Figure A. \FIGURE[scale=1]{}{../s/c0001/s0020/e0180/}{A}{Gehman dendrite} %PP: put, please a picture of the Gehman dendrite here. The Gehman dendrite has the following properties. \rostere \item The set $R(G)$ of all ramification points of $G$ is discrete. \item $E(G) = \cl(R(G)) \setminus R(G)$. \item Each dendrite with an uncountable set of its end points contains a homeomorphic copy of the Gehman dendrite, \cite[Proposition 6.8, p. 16]{Arevalo+2001a}. %A space $X$ is said to have the \g{periodic-recurrent property} %\g{periodic-recurrent property} provided that for each mapping $f: X %\to X$ the closures of the sets of periodic and of recurrent points of $f$ %are equal (see \cite[Definition 1.2, p. 109]{Charatonik+1997b}). The following results %are known. \item If a continuum contains the Gehman dendrite, then it does not have the \g{periodic-recurrent property}, \cite[Theorem 3.3, p. 136]{Charatonik1998a}. \item A dendrite $X$ contains the Gehman dendrite if and only if $X$ does not have the periodic-recurrent property, \cite[Theorem 2, p. 222]{Illanes1998a}. \rosteree \STORY{../s/c0001/s0020/e0180} %%%%%%%%% %%%%%%%%% END %%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% START %%%%%%%%% %%%%%%%%% /c0001/s0020/e0190 %%%%%%%%% %%%%%%%%% \subsection{ {Modifications of the Gehman dendrite} } \setcounter{equation}{0} The concept of the \g{Gehman dendrite} has been generalized in \cite[Section 4, p. 4-10]{Arevalo+2001a} as follows. \rostere \item For each natural number $n \ge 3$ there exists topologically unique dendrite $G_n$ \g{dendrite $G_n$} such that \rostere \item $\mathrm{ord}\, (p, G_n) = n$ for each point $p \in R(G_n)$; \item $E(G_n)$ is homeomorphic to the Cantor ternary set. \rosteree (Note that $G_3$ is just the Gehman dendrite $G$.) \item There exists a \es{dendrite $G_{\omega}$}{dendrite Gomega} such that \rostere \item $\ord(p, G_{\omega})$ is finite for each point $p \in G_{\omega}$; \item $E(G_{\omega})$ is homeomorphic to the Cantor ternary set; \item for each natural number $n$ and for each maximal arc $A$ contained in $G_{\omega}$ there is a point $q \in A$ such that $\ord(q, G_{\omega}) \ge n$. \rosteree See Figure A. \FIGURE[scale=1]{}{../s/c0001/s0020/e0190/}{A}{$G_4$} The dendrites $G_n$ and $G_{\omega}$ have the following universality properties. \item For each natural number $n \ge 3$ the dendrite $G_n$ is \gs{universal}{universal dendrite} for the class of all dendrites $X$ such that $\cl E(X) = E(X)$ and that $\ord(p,X) \le n$ for each point $p \in X$. \item Each dendrite $G_{\omega}$ is universal for the class of all dendrites with the closed set of end points. \rosteree \STORY{../s/c0001/s0020/e0190} %%%%%%%%% %%%%%%%%% END %%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% START %%%%%%%%% %%%%%%%%% /c0001/s0020/e0200 %%%%%%%%% %%%%%%%%% \subsection{ {Mapping hierarchy of dendrites} } \setcounter{equation}{0} A classification of topological spaces from the standpoint of the theory of mappings (called also a \g{mapping hierarchy of spaces}) is defined in \cite{Borsuk1959a}, and it is investigated for dendrites in \cite{Charatonik+1994a}. Some questions asked there have already been answered. We recall some examples related to this subject. \rostere \item There exist two dendrites $X$ and $Y$ such that $\{X, Y\}$ has no infimum and no supremum with respect to the class of \gs{monotone mappings}{monotone mapping} (see \cite[Example 5.45, p. 18]{Charatonik+1994a}). \item There exist dendrites $X$ and $Y$, both admitting \gs{open mappings}{open mapping} onto arcs, such that $Y \subset X$ and there is no open mapping from $X$ onto $Y$ (see \cite[Example 6.67, p. 34]{Charatonik+1994a}) \par Indeed, in an arm $A$ of a simple triod $Y$ fix a sequence $\{p_n\}$ of points of $A$ converging to the center of the triod, and take a sequence $\{A_n\}$ of arcs such that $Y \cap A_n = \{p_n\}$ and that $\lim \diam A_n = 0$. Then $X = Y \cup \bigcup\{A_n: n \in \mathbb N\}$ is the needed dendrite. \item There exists a dendrite which does not admit any \g{open mapping} onto an arc and which is the union of two subdendrites admitting open mappings onto an arc (see \cite[Example 6.69, p. 34]{Charatonik+1994a}). %\item There exists an openly minimal dendrite that has two topologically %different open images, see \cite[Proposition 3.5, p. 233]{Opela+2000a}. Openly %minimal dendrites having a prescribed finite number of different open %images are studied in \cite{Jelinek+2001a}. \item There is a bounded chain of dendrites (ordered with respect to open mappings) that has no supremum (see \cite{Podbrdsky+2001a}). \item There is in the plane an uncountable family of dendrites every two members of which are \gs{incomparable by $r$-mappings}{spaces incomparable by a class of mappings} (see \cite{Sieklucki1959a}). \item There is a family of dendrites ordered with respect to $r$-mappings similarly to the segment (see \cite{Sieklucki1961a}). \rosteree \STORY{../s/c0001/s0020/e0200} %%%%%%%%% %%%%%%%%% END %%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{ Cyclic examples of locally connected continua} \begin{itemize} \item [(a)] Claytor examples of nonplanable curves \item [(b)] Menger compacta, Sierpi\'nski universal plane curve, Menger universal curve; homogeneity of these continua \item [(c)] Other examples \end{itemize} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% START %%%%%%%%% %%%%%%%%% /c0001/s0030/e0040 %%%%%%%%% %%%%%%%%% \subsection{ {Menger universal continua} } \setcounter{equation}{0} \label{ex-Menger} The Menger universal continuum $M^m_n$, for $1\le n0$, there is $\delta>0$ such that if $Z'\subset M^{2n+1}_n$ is a Z-set and $h:Z\to Z'$ is a $\delta$-homeomorphism, then $h$ extends to an \gs{$\epsilon$-homeomorphism}{epsilon-homeomorphism} $h^*:M^{2n+1}_n\to M^{2n+1}_n$ such that $h^*|M^{2n+1}_n\setminus U=\operatorname{identity}$ \cite[Theorem 3.1.1, p. 65]{Bestvina1988a}. \item\label{ex-Menger-Menger4b} For every $\epsilon>0$, there exists $\delta>0$ such that if $Z,Z'$ are Z-sets\g{Z-set} in $ M^{2n+1}_n$ and $h:Z\to Z'$ is a $\delta$-homeomorphism, then there is an $\epsilon$-homeomorphism $h^*: M^{2n+1}_n\to M^{2n+1}_n$ extending $h$ \cite[Theorem 3.1.3, p. 71]{Bestvina1988a}. \item Every homeomorphism between Z-sets in $M^{2n+1}_n$ extends to an autohomeomorphism of $M^{2n+1}_n$ \cite[Corollary 3.1.5, p. 72]{Bestvina1988a}. \rosteree It follows that $M^{2n+1}_n$ is \g{strongly locally homogeneous} and \g{countably dense homogeneous}. \item If $m<2n+1$, then $M^m_n$ is not homogeneous \cite{Lewis1987a} (for $m\ge 2n+1$ the homogeneity of $M^m_n$ follows from the property above). \item $M^{2n+1}_n\times X$ is not 2-homogeneous for an arbitrary continuum $X$ \cite{Kuperberg+1995a}. \item\label{ex-Menger-autohomeo} The groups of autohomeomorphisms of $M^{2n+1}_n$ and $M^{n+1}_n$ are Polish and one-dimensional (see \cite[Corollary 6]{Oversteegen+1994a} and Property~\ref{ex-Menger-Menger2}). The group is totally disconnected for $M^{2n+1}_n$ (see \cite[Theorem 1.3]{Brechner1966a} for $n=1$ and its natural extension for any $n$). \item The Polish group of autohomeomorphisms of $M^m_n$ is at most one-dimensional \cite{Oversteegen+1994a}. \item The group of autohomeomorphisms of $M^{2n+1}_n$ is \gs{simple}{simple group} (see \cite[Theorem 3.2.4]{Chigogidze+1995a}). \item Every autohomeomorphism of $M^{2n+1}_n$ is a composition of two homeomorphisms, each of which is the identity on some nonempty open set \cite{Sakai1994a}. \item For each $t\in[n,2n+1]$, there is a copy of $M^{2n+1}_n\subset \mathbb R^{2n+1}$ with the \g{Hausdorff dimension} $t$ \cite[Theorem 4.2.6]{Chigogidze+1995a}. \item Every Z-set in $M^{2n+1}_n$ is the fixed point set of some autohomeomorphisms of $M^{2n+1}_n$ \cite{Sakai1997a}. \item Any $C^{n-1}$ and $LC^{n-1}$-continuum is the image of $M^{2n+1}_n$ under a \gs{$UV^{n-1}$-map}{UVn-map} \cite[Theorem 5.1.8, p. 95]{Bestvina1988a}. \item If a Polish space $X$ contains a topological copy $M$ of $M^{2n+1}_n$ or $M^{n+1}_n$, then the space of all copies of $M$ in $X$ with the \g{Hausdorff metric} is not a $G_{\delta\sigma}$-set \cite{KrupskiXXXXb}. \rosteree \STORY{../s/c0001/s0030/e0040} %%%%%%%%% %%%%%%%%% END %%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% START %%%%%%%%% %%%%%%%%% /c0001/s0030/e0050 %%%%%%%%% %%%%%%%%% \subsection{ {Sierpi\'nski carpet} } \setcounter{equation}{0} The \es{Sierpi\'nski universal plane curve}{Sierpinski universal plane curve} or the \es{Sierpi\'nski carpet}{Sierpinski carpet} \cite{Sierpinski1916a} $M^2_1$ is a well known \g{fractal} obtained as the set remaining when one begins with the unit square $I^2$ and applies the operation of dividing it into 9 congruent squares and deleting the interior of the central one, then repeats this operation on each of the surviving 8 squares, and so on. See Figure A. \FIGURE[scale=1]{}{../s/c0001/s0030/e0050/}{A}{Sierpi\'nski carpet} %Insert Figure 3.3.2 \GIF[scale=1]{}{../s/c0001/s0030/e0050/}{AA}{Sierpi\'nski carpet - an animation } \rostere \item $M^2_1$ is \g{universal} in the class of all at most one-dimensional subsets of the plane (equivalently, of all boundary subsets of the plane) \cite{Sierpinski1916a}, \cite{Sierpinski1922a}. \item The following statements are equivalent: \rostere \item $X$ is homeomorphic to $M^2_1$; \item $X$ is a \g{locally connected} plane \g{curve} that contains no \gs{local cut points}{local cut point}; \item $X$ is a continuum embeddable in the plane in such a way that $\mathbb R^2\setminus X$ has infinitely many components $C_1, C_2,\dots$ such that $\diam C_i\to 0$, $\bd C_i\cap \bd C_j=\emptyset$ for $i\ne j$, $\bd C_i$ is a \g{simple closed curve} for each $i$ and $\bigcup_{i=1}^\infty \bd C_i$ is dense in $X$ \cite{Whyburn1958a}. \rosteree \item A complete metric space $X$ contains a topological copy of $M^2_1$ if and only if $X$ contains a subset with the \g{bypass property} \cite{Prajs1998a}. \item The group of all autohomeomorphisms of $M^2_1$ has exactly two orbits: one of them is the union of all \gs{simple closed curves}{simple closed curve} which are the boundaries of complementary domains of $M^2_1$ \cite{Krasinkiewicz1969a}. The group is a Polish topological group which is totally disconnected and one-dimensional (see \cite[Theorem 1.2]{Brechner1966a} and Property~\ref{ex-Menger-autohomeo} in \ref{ex-Menger}). \item Any homeomorphism between Cartesian products of copies of the Sierpi\'nski carpet is \gs{factor preserving}{factor preserving homeomorphism} \cite{Kennedy1980a}. Consequently, no such product is \g{homogeneous}. \item The Sierpi\'nski carpet can be \gs{continuously decomposed}{continuous decomposition} into \gs{pseudo-arcs}{pseudo-arc} such that the decomposition space is homeomorphic to the carpet \cite[Corollary 18]{Prajs1998b}, \cite{Seaquist1995a}. In fact, $M^2_1$ is the only planar locally connected curve admitting such a decomposition \cite[Corollary 18]{Prajs1998b}. \item The Sierpi\'nski carpet is \gs{homogeneous with respect to} {homogeneous with respect to a class of mappings} \gs{monotone open mappings}{monotone open mapping} \cite[Corollary 24]{Prajs1998b}, \cite{Seaquist1999a}. Moreover, every continuum which is locally homeomorphic to $M^2_1$ (i. e., $M^2_1$-manifold) is homogeneous with respect to monotone open mappings \cite[Theorem 23]{Prajs1998b}. \item $M^2_1$ is homogeneous with respect to the class of \gs{simple mappings}{simple mapping} \cite{Charatonik1984a}. \item If $C$ is a curve, then the set of all mappings $f:C\to \mathbb R^2$ such that $f(C)$ is homeomorphic to $M^2_1$ is a \gs{residual}{residual set} subset of the space $(\mathbb R^2)^C$ of all mappings of $C$ into $\mathbb R^2$ with the uniform convergence metric. If $C(X)$ is the \gs{hyperspace}{hyperspace of subcontinua} of all subcontinua of a compact space $X$ and $C_1(X)$ its subspace of all curves, then the set $$\{\,(C,f)\in C(X)\times (\mathbb R^2)^X: \text{$f(C)$ is homeomorphic to $M^2_1$}\,\}$$ is residual in $C(X)\times (\mathbb R^2)^X$; in other words, almost all mappings in $(\mathbb R^2)^X$ map almost all curves in $X$ onto copies of $M^2_1$, where "almost all" means all with except of a subset of the first category in corresponding spaces \cite{Mazurkiewicz1938a}. \item If a locally compact space $X$ contains a topological copy of the Sierpi\'nski carpet, then the space of all copies of the Sierpi\'nski carpets in $X$ with the Hausdorff metric\g{Hausdorff metric} is a true absolute $F_{\sigma\delta}$-set \cite{KrupskiXXXXb}. \rosteree \STORY{../s/c0001/s0030/e0050} %%%%%%%%% %%%%%%%%% END %%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% START %%%%%%%%% %%%%%%%%% /c0001/s0030/e0060 %%%%%%%%% %%%%%%%%% \subsection{ {Menger universal curve} } \setcounter{equation}{0} \label{ex-Menger curve} The \e{Menger universal curve} $M^3_1$ is the subset of the unit cube $I^3$ whose projections onto faces of the cube are the \gs{Sierpi\'nski carpets}{Sirpinski carpet}, i.e., $$M^3_1=\{\,(x,y,z)\in I^3: (x,y)\in M^2_1,\quad (y,z)\in M^2_1, \quad (x,z)\in M^2_1\,\}$$ \cite{Menger1926a}. Other descriptions can be found in \ref{ex-Menger} (see also \cite[pp. 5--6, 8--9]{Mayer+1986a}). See Figure A. \FIGURE[scale=1]{}{../s/c0001/s0030/e0060/}{A}{Menger universal curve} \GIF[scale=1]{}{../s/c0001/s0030/e0060/}{AA}{Menger universal curve - an animation} %insert Figure 3.3.3 \rostere \item The following statements are equivalent: \rostere \item $X$ is homeomorphic to $M^3_1$; \item $X$ is a locally connected curve with no local cut points and no planar open nonempty subsets \cite[Theorem XII, p. 13]{Anderson1958a}; \item $X$ is a \g{homogeneous} locally connected \g{curve} different from a \g{simple close curve} \cite[Theorem XIII, p. 14]{Anderson1958a}; \item $X$ is a locally connected curve with the \g{disjoint arcs property} (see Property~\ref{ex-Menger-Menger3} in \ref{ex-Menger}); \item $X$ is a locally connected curve and each arc in $X$ is \g{approximately non-locally-separating arc} and has empty interior in $X$ \cite[Theorem 3, p. 86]{Krupski+XXXXa}. \rosteree \item $M^3_1$ is \gs{universal}{universal space} in the class of all metric separable spaces of dimension $\le 1$ \cite{Menger1926a}. \item Z-sets\g{Z-set} in $M^3_1$ coincide with \gs{non-locally-separating closed subsets}{non-locally-separating set} of $M^3_1$. If $Z$ is a separable metric space of dimension $\le 1$, then $Z$ can be embedded as a non-locally-separating subset of $M^3_1$ \cite[Theorem 6.1, p. 42]{Mayer+1986a}. \item If $Z$ is a closed subset of a metric space $X$, $\dim(X\setminus Z)\le 1$ and $f:Z\to M^3_1$ is a continuous mapping with non-locally separating image $f(Z)$, then $f$ can be extended to a map $g:X\to M^3_1$ such that $g|X\setminus Z$ is an embedding into $M^3_1\setminus f(Z)$ \cite[Theorem 6.4, p. 44]{Mayer+1986a}. \item Every continuous surjection $h$ between non-locally separating closed subsets $Z,Z'$ of $ M^3_1$ extends to a mapping $h^*$ of $M^3_1$ such that $h^*|M^3_1\setminus Z$ is a homeomorphism onto $M^3_1\setminus Z'$ \cite[Corollary 6.5, p. 44]{Mayer+1986a}; moreover, for every $\epsilon>0$ there exists $\delta>0$ such that if $h$ is a $\delta$-homeomorphism, then $h^*$ can be taken as an $\epsilon$-homeomorphism (see Property~\ref{ex-Menger-Menger4b} in \ref{ex-Menger}). \item For each locally connected continuum $X$, there exist open surjections $f:M^3_1\to X$ and $g:M^3_1\to X$ such that $f^{-1}(x)$ is homeomorphic to $M^3_1$ and $g^{-1}(x)$ is a Cantor set, for any $x\in X$ \cite{Wilson1972a}. \item Any compact 0-dimensional group $G$ acts freely on $M^3_1$ so that the orbit space $M^3_1/G$ is homeomorphic to $M^3_1$ \cite[Theorem 1]{Anderson1957a}. \item If a locally compact space $X$ contains a topological copy of $M^3_1$, then the space of all copies of $M^3_1$ in $X$ with the Hausdorff metric\g{Hausdorff metric} is a true absolute $F_{\sigma\delta}$-set \cite{KrupskiXXXXb}. \rosteree \STORY{../s/c0001/s0030/e0060} %%%%%%%%% %%%%%%%%% END %%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{ 2-dimensional locally connected continua } \begin{itemize} \item [(a)] Surfaces, polyhedra \item [(b)] Pontryagin surfaces, Pontryagin disk (sphere) \item [(c)] 2-dimensional AR containing no disk \item [(d)] Borsuk's hat \item [(e)] Alexander horned sphere \item [(f)] Bing dogbone space \item [(g)] Other examples \end{itemize} \section{ Higher dimensional locally connected continua } \begin{itemize} \item [(a)] $n$-dimensional manifolds \item [(b)] Hilbert cube manifolds \item [(c)] Other important examples \end{itemize} \section{ Variations of local connectedness} \begin{itemize} \item [(a)] Hereditary local connectedness \item [(b)] Connectedness im kleinen \item [(c)] Semi-local connectedness \end{itemize} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% %%%%%%%%% START of CHAPTER %%%%%%%%% /c0002 %%%%%%%%% %%%%%%%%% \chapter{ JOINING PROPERTIES} \section{ Cut points and separating points (also local)} \section{ Arcwise connectedness; arc components} \section{ Continuum chainability} \section{ Various kinds of connectedness ($\lambda$-connectedness, $\delta$-connectedness, etc.)} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% %%%%%%%%% START of CHAPTER %%%%%%%%% /c0003 %%%%%%%%% %%%%%%%%% \chapter{ DECOMPOSABLE AND INDECOMPOSABLE CONTINUA} \section{ Decomposable continua (also hereditary)} \section{ Indecomposable continua (also hereditary)} \begin{itemize} \item [(a)] Lakes of Wada \item [(b)] Knaster type continua; Brouwer-Janiszewski-Knaster (buckethandle) continuum \item [(c)] Other simple examples \item [(d)] The pseudo-arc \end{itemize} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% START %%%%%%%%% %%%%%%%%% /c0003/s0020/e0050 %%%%%%%%% %%%%%%%%% \subsection{ {Knaster continua} } \setcounter{equation}{0} By a \e{Knaster continuum} is meant a continuum homeomorphic to the inverse limit $\varprojlim(I_n,f_n)$ of a sequence of unit intervals $I_n=I$ with open, non-homeomorphic bonding maps $f_n$. Denote by $\mathcal K$ the class of all Knaster continua. The most popular examples of Knaster continua is the \e{buckethandle} (sometimes called by dynamists the \e{horseshoe} ) $B_1$, where $f_n(t)=1-|2t-1|$, and the \e{double buckethandle} $B_2$ with $$ f_n(t)=\begin{cases} 3t& \text{if \ $0\le t\le\frac13$},\\ 2-3t& \text{if \ $\frac13\le t\le \frac23$},\\ 3t-2& \text{if \ $\frac23\le t\le 1$}. \end{cases} $$ Nice geometric models of $B_1$ and $B_2$ can be described in the following way (see \cite[pp. 204--205]{Kuratowski1968a}). If $C$ denotes the standard Cantor ternary set in $I$, then $B_1$ is homeomorphic to the union of all semi-circles $$\{\,(x,y)\in\mathbb R^2: (x-\frac12)^2+y^2=r^2, y\ge 0\,\},$$ where $r+\frac12\in C$, and all semi-circles $$\{\,(x,y): (x-\frac5{2\cdot3^n})^2+y^2=r^2, y\le 0\,\},$$ for $n\in\mathbb N$, where $r+\frac5{2\cdot3^n}\in C\cap [\frac2{3^n},\frac1{3^{n-1}}]$. See Figure A. \FIGURE[scale=1]{}{../s/c0003/s0020/e0050/}{A}{buckethandle} %Figure 14.1.6 If $E$ is the quintary Cantor set in $I$ (real numbers in $I$ which can be written in the enumeration system at base 5 without digits 1 and 3), then $B_2$ is homeomorphic to the union of all semi-circles $$\{\,(x,y): (x-\frac7{10\cdot5^n})^2+y^2=r^2, y\le0\,\},$$ for $n\in\mathbb N$ and $r+\frac7{10\cdot5^n}\in E\cap [\frac2{5^{n+1}},\frac1{5^n}]$ and all semi-circles $$\{\,(x,y):(x-(1-\frac7{10\cdot5^n}))^2+y^2=r^2, y\ge 0\,\},$$ for $n\ge 0$ and $\frac2{5^{n+1}}\le\frac7{10\cdot5^n}-r\le\frac1{5^n}$. See Figure B. \FIGURE[scale=1]{}{../s/c0003/s0020/e0050/}{B}{double buckethandle} %Figure 13.1.8 \GIF[scale=1]{}{../s/c0003/s0020/e0050/}{BB}{double buckethandle} \rostere \item If $w_p:I\to I$ is the map such that $w_p(i/p)=0$ for even $i$, $w_k(i/p)=1$ for odd $i$, where $i$ is an integer between 0 and $p\in\mathbb N$, and $w_p$ is linear on each interval $[i/p,(i+1)/p]$, then $\mathcal K$ is equal to the family of all continua homeomorphic to inverse limits $\varprojlim(I_n,f_n)$, where, for each $n$, $I_n=I$ and $f_n=w_{p_n}$ for some $p_n$ \cite{Rogers1970a}. So, each $K\in\mathcal K$ is determined by a sequence $\mathbf{p} =(p_1,p_2,\dots)$ such that $K= \varprojlim(I_n,w_{p_n})$. We will denote such $K$ by $K(\mathbf{p})$ and call it a \es{$\mathbf p$-adic Knaster continuum}{p-adic Knaster continuum} . Without loss of generality, one can assume that all $p_n$'s are prime. \item Each $\mathbf p$-adic Knaster continuum $K$ is homeomorphic to the quotient of the \gs{$\mathbf{p}$-adic solenoid}{p-adic solenoid} $\Sigma(\mathbf{p})$ under the relation $$a\sim b\Leftrightarrow a=b\quad \text{or}\quad a=b^{-1}, $$ where $a,b\in \Sigma(\mathbf{p})$ and $b^{-1}$ is the group inverse of $b$. A point of $K$ is an \g{end point} of $K$ if and only if it is an image under the quotient map of the neutral element or the element of order 2 (if such exists) of the corresponding solenoid (see, e.g., \cite[p. 43]{Krupski1984b}). \item The class $\mathcal K$ of all Knaster continua is equal to the class of all \g{arc-like} continua with one or two end points having the \g{property of Kelley} and arcs as proper non-degenerate subcontinua and which themselves are not arcs \cite{Krupski1984b}. In particular, all of them are \g{indecomposable}. \item There are $2^{\aleph_0}$ mutually non-homeomorphic members of $\mathcal K$ \cite{Watkins1982a}; in fact, there is a subfamily of $\mathcal K$ of cardinality $2^{\aleph_0}$ no member of which is an open image of another one \cite{Debski1985a}. \item Any open map between two Knaster continua $\varprojlim(I_n,f_n)$, $\varprojlim(I_n,g_n)$ is the uniform limit of open maps which are induced by maps $(I_n,f_n)\to (I_n,g_n)$ between the inverse sequences \cite[Theorem 4.8, p. 145]{Eberhart+1999a}. \item If $K\in \mathcal K$, then every indecomposable continuum can be mapped onto $K$ \cite[Corollary, p. 455]{Rogers1970a}. Moreover, if $X$ is an indecomposable (Hausdorff) continuum, $a,b\in X$ and $a\ne b$ ($a$ and $b$ belong to different \gs{composants}{composant} of $X$), then there is a continuous surjection $f:X\to B_1$ such that $f(a)\ne f(b)$ ($f(a)$ and $f(b)$ belong to different composants of $B_1$, resp.). It follows that any indecomposable continuum embeds in the Cartesian product of copies of $B_1$ \cite[Corollaries 1 and 2, p. 305]{Bellamy1973a}. \item If $K_1, K_2\in \mathcal K$ then there are $2^{\aleph_0}$ distinct \gs{homotopy types}{homotopy type} of maps of $K_1$ onto $K_2$ that map an end point of $K_1$ onto an end point of $K_2$ \cite{Minc1999a}. \item If $K\in \mathcal K$ and $f$ is a \g{monotone map} of $K$, then $f(K)$ is homeomorphic to $K$ \cite{Krupski1984b}. \item There is no exactly 2-to-1 map defined on a Knaster continuum \cite{Debski1992a}. \item Knaster continua are \gs{absolute retracts}{absolute retract} for the class of all \g{hereditarily unicoherent} continua, i.e., if $K\in\mathcal K$, $X$ is a hereditarily unicoherent continuum and $K\subset X$, then $K$ is a retract of $X$. Equivalently, if $A$ is a closed subset of a hereditarily unicoherent continuum $X$, then every map $f$ from $A$ onto a Knaster continuum can be extended over $X$ (\cite{Mackowiak1984a} for $K=B_1$ and $X$ a Hausdorff hereditarily unicoherent continuum and \cite{Charatonik+XXXXa} for arbitrary $K\in\mathcal K$). \item Any two points of a plane continuum $X$ can be joined by a \g{hereditarily decomposable} subcontinuum if and only if $X$ cannot be mapped onto $B_1$ \cite[Theorem 2, p. 133]{Hagopian1974a}. \item Every autohomeomorphism of the buckethandle $B_1$ is \g{isotopic} to some iterate of the shift homeomorphism $f:B_1\to B_1, f(x_1,x_2,\dots)=(x_2,x_3,\dots)$ \cite[Theorem 4.4, p. 204]{Aarts+1991a}. Moreover, any autohomeomorphism $f$ of a Knaster continuum $\varprojlim(I_n,w_{p_n})$ is isotopic to a standard homeomorphism $g$ (defined as a map induced by a map $(I_n,w_{p_n})\to (I_n,w_{p_n})$ of inverse systems); maps $f$ and $g$ have the same topological entropy and if the entropy is positive, then they are \g{semi-conjugate} \cite[Theorem 4, p. 271]{Kwapisz2001a}. \item If $e$ is the end point of $B_1$ and $S, T$ are composants of $B_1$ which do not contain $e$, then there exists a continuous injection $h:B_1\setminus\{e\}\to B_1\setminus\{e\}$ such that $h(S)=T$ \cite[Theorem 5.1]{Aarts+1991a}. Any two composants of $B_1$ which do not contain $e$ are homeomorphic \cite{Bandt1994a}. \item Every autohomeomorphism of the buckethandle continuum has at least two \gs{fixed points}{fixed point} \cite{Aarts+1998a}. \item If $A$ is a Borel subset of $K\in\mathcal K$ and $A$ is the union of a family of composants of $K$, then $A$ is \g{meager} or \g{comeager} \cite{Emeryk1980a}, \cite{Krasinkiewicz1974a}. \item Each composant $C$ of $B_1$ different from the one containing the end point is internal, i.e., every continuum $L\subset \mathbb R^2$ intersecting both $C$ and $\mathbb R^2\setminus B_1$ must intersect all composants of $B_1$ \cite[Theorem, p. 261]{Krasinkiewicz1974b}. \item Any Knaster continuum $K=\varprojlim(I_n,f_n)$, where $f_m=f_n$ for each $m,n$, is homeomorphic to the \g{attracting set} of a homeomorphism $h:D^2\to h(D^2)\subset D^2$ of the plane closed disk \cite{Barge1986a}. If $f_n(t)=1-|2t-1|$ for all $n$ (i.e., $K=B_1$), then $h$ can be defined as a diffeomorphism which is called a \h{horse-shoe map}---it was first described by S. Smale in \cite{Smale1965a} (see also \cite{Barge1988a}). See Figure C. \FIGURE[scale=1]{}{../s/c0003/s0020/e0050/}{C}{the horse-shoe map $f$ maps the stadium shaped region $D$ ($A \cup R \cup B$) in such a way, that it shrinks $R$ vertically, stretches $R$ horizontally, contracts the semicircles then folds the space once and places this result into itself so that $f(A)$ and $f(B)$ are in the interior of $A$ and $f(R)$ is in the interior of $D$.} %Include a picture of the horseshoe map as, e.g., in %J. A. Kennedy, A brief history of indecomposable continua, Continua with the Houston problem book, Dekker, 1995, p. 117, Figure 6. %or in R. L. Devaney, An Introduction to Chaotic Dynamical Systems, 2nd ed., Addison-Wesley, 1989, %Figure 3.2, p. 182 \item The buckethandle $B_1$ and all Knaster continua with two end points admit no \g{mean} \cite{IllanesXXXXa}, \cite[Theorem 2.2, p. 99]{Kawamura+1996a}. \item The \g{hyperspace} $C(K)$ of all subcontinua of any Knaster continuum $K$ is homeomorphic to the \g{cone} $Cone(K)$ over $K$ by a homeomorphism $h:C(K)\to Cone(K)$ which sends $K$ onto the vertex of $Cone(K)$ and the set of all singletons of $K$ onto the base of the cone \cite[Corollary 12, p. 639]{Dilks+1981a}. \rosteree \STORY{../s/c0003/s0020/e0050} %%%%%%%%% %%%%%%%%% END %%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{ Unicoherent continua (also hereditary)} \section{ Discoherent continua} \section{ Multicoherent continua} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% %%%%%%%%% START of CHAPTER %%%%%%%%% /c0004 %%%%%%%%% %%%%%%%%% \chapter{ IRREDUCIBLE CONTINUA} \section{ Elementary examples} \begin{itemize} \item [(a)] An arc, Sin Curve \item [(b)] Other examples (Cantor Organ, Cantor Accordion, etc.) \end{itemize} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% START %%%%%%%%% %%%%%%%%% /c0004/s0010/e0010 %%%%%%%%% %%%%%%%%% \subsection{ {Sin curve} } \setcounter{equation}{0} The \eS{(topologist's)}{topologist's sin curve} \e{sin curve} $S$ is defined by $$S=\{\,(x,y)\in \mathbb R^2: y=\sin\frac1x, 00$ there exist a closed-open subset $D$ of $C$ and a positive integer $n$ such that $\{f\sp j(D)\colon j=1,\cdots,n\}$ is a cover of $C$ consisting of pairwise disjoint subsets of $C$ with diameters less than $\epsilon$ \cite{Gutek1980a}. \item Each solenoid $\Sigma(\mathbf p)$ is an Abelian \g{topological group} with a group operation $(z_1,z_2,\dots)\cdot (z'_1,z'_2,\dots)=(z_1\cdot z'_1,z_2\cdot z'_2,\dots)$ and the neutral element $e=(1,1,\dots)$. \item Either of the following conditions is equivalent for a nondegenerate continuum $X$ different from a simple closed curve to be a solenoid. \rostere \item $X$ is homeomorhic to a \g{one-dimensional} topological group \cite{Hewitt+1963a}; \item $X$ is \g{indecomposable} and is homeomorphic to a topological group \cite[Theorem 8.6.18]{Chigogidze1996a}; \item $X$ is \g{circle-like}, has the \g{property of Kelley} and contains no \g{local end point} \cite[Theorem (4.3)]{Krupski1984c}; \item $X$ is \g{circle-like}, has the \g{property of Kelley}, each proper nondegenerate subcontinuum of $X$ is an arc and $X$ has no \g{end points}{end point}; \item $X$ is circle-like, has the property of Kelley and has an open cover by Cantor bundles of open arcs (i.e., sets homeomorphic to the product $C\times(0,1)$ of the Cantor set $C$ and the open interval $(0,1)$) \cite{Krupski1982a}; \item $X$ is \g{homogeneous}, contains no proper, nondegenerate, \gs{terminal}{terminal subcontinuum} subcontinua and sufficiently small subcontinua of $X$ are not \gs{$\infty$-ods}{infty-od} \cite[Theorem 3.1]{Krupski1995a}; \item $X$ is a homogeneous curve containing an open subset $U$ such that some component of $U$ does not have the \g{disjoint arcs property} \cite[p. 166]{Krupski1995a}; \item $X$ is a homogeneous \gs{finitely cyclic}{finitely cyclic curve} (or, equivalently, \gs{$k$-junctioned}{k-junctioned curve}) curve that is not tree-like and contains no nondegenerate, proper, terminal subcontinua \cite{Krupski+XXXXb}, \cite{Duda+1991a}. \item $X$ is \g{openly homogeneous} and sufficiently small subcontinua of $X$ are arcs \cite{Prajs1989a}; \rosteree \item Solenoid $\Sigma(\mathbf q)$ is a continuous image of $\Sigma(\mathbf p)$ if and only if the sequence $\mathbf q=(q_1,q_2,\dots)$ is a \g{factorant} of sequence $\mathbf p=(p_1,p_2,\dots)$, i.e., there exists $i$ such that for each $j\ge i$ there is $k$ such that $q_i\cdot \dots\cdot q_j$ is a factor of $p_1\cdot\dots\cdot p_k$. Two solenoids are homeomorphic if and only if each of them is a continuous image of another \cite{Cook1967a}, \cite[Satz 8, p. 122]{Dantzig1930a}. There is a family of solenoids of cardinality $2^{\aleph_0}$ such that no member of the family is a continuous image of another. \item Each monotone\g{monotone map} image of a solenoid $X$ is homeomorphic to $X$ \cite[Theorem 5]{Krupski1984b}. Each open map transforms $X$ onto a solenoid or onto an arc-like continuum with the property of Kelley and with arcs as proper nondegenerate subcontinua; if the map is a local homeomorphism, then its image is a solenoid \cite{Krupski1984a}. \item The composant of a solenoid $\Sigma$ containing $e$ is a one-parameter topological subgroup of $\Sigma$, i.e. it is a one-to-one continuous homomorphic image of the additive group of the reals. \item Any two \gs{composants}{composant} of any two solenoids are homeomorphic \cite{Man1995a}. %\item %Composants of solenoids are not planable. \item No solenoid can be mapped onto a \g{strongly self-entwined continuum}. In particular, it cannot be mapped onto a circle-like plane continuum which is a common part of a descending sequence of circular chains $C_i$ such that $C_{i+1}$ circles $n$ times in $C_i$ clockwisely and then $n-1$ times counter-clockwisely and the first link of $C_i$ contains the closure of the first link of $C_{i+1}$ \cite{Rogers1971b}. \item No \gs{movable}{movable continuum} continuum (in particular no continuum lying in a surface or a \gs{tree-like}{tree-like continuum} continuum) can be continuously mapped onto a solenoid. Alternatively, if the first Alexander-\v Cech cohomology group of a continuum $X$ is finitely divisible\g{finitely divisible group}, then $X$ cannot be mapped onto a solenoid \cite[Remark, p. 46, 4.1, 4.9., 5.1]{Krasinkiewicz1976a}, \cite[Corollary 7.3]{Krasinkiewicz1978a}, \cite{Rogers1975a}. \item Every nonplanar, circle-like continuum has the \g{shape} of a solenoid \cite[remark, p. 46]{Krasinkiewicz1976a}. Two solenoids have the same shape if and only if they are homeomorphic \cite{Godlewski1970a}. \item Any autohomeomorphism $f$ of $\Sigma(\mathbf p)$ is isotopic to a homeomorphism $g$ which is induced by a map $(S^1,f_n)\to (S^1,f_n)$ of the inverse sequences which define $\Sigma(\mathbf p)$ ($g$ can be a group translation, the involution, a power map or its inverse, or compositions of these maps). Maps $f$ and $g$ have equal the \gs{topological entropies}{topological entropy} and are \g{semi-conjugate} if the entropy is positive \cite[Theorems 1--3, pp. 252--253]{Kwapisz2001a}, \cite[Satz 9, p. 125]{Dantzig1930a}. The topological group of all autohomeomorphisms (with the compact-open topology) of a solenoid $\Sigma$ is homeomorphic (but not isomorphic) to the the product $\Sigma\times l_2\times Aut(\Sigma)$, where $l_2$ is the Hilbert space and the group $Aut(\Sigma)$ of all topological group automorphisms of $\Sigma$ is equipped with the discrete topology and it is equal to $\mathbb Z_2$, or $\mathbb Z_2\times\mathbb Z^n$, or $\mathbb Z_2\oplus_{i=1}^\infty\mathbb Z$ \cite[Theorems 3.1 and 2.4]{Keesling1972a}. \item If the spaces of all autohomeomorphisms of two solenoids are homeomorphic, then the solenoids are isomorphic as topological groups \cite[Corollary 3.9]{Keesling1972a}. \item Any map $f:\Sigma(\mathbf p)\to \Sigma(\mathbf q)$ is, for every $\epsilon>0$, $\epsilon$-homotopic to a map induced by a map $(S^1,f_n)\to (S^1,g_n)$ between inverse sequences defining the corresponding solenoids \cite{Rogers+1971a}. \item A $\mathbf p$-adic solenoid admits a \g{mean} if and only if infinitely many numbers in the sequence $\mathbf p$ equal 2 \cite{KrupskiXXXXa}. The same condition is equivalent to the non-existence of exactly 2-to-1 map defined on the solenoid \cite{Debski1992a}. \item The \gs{hyperspace}{hyperspace of subcontinua} of all subcontinua of any solenoid $\Sigma$ is homeomorphic the cone over $\Sigma$ \cite{Rogers1971a}, \cite[p. 202]{Nadler1991a}. \item The family of all solenoids in the cube $I^3$ (as a subset of the hyperspace $C(I^3)$) is Borel and not $G_{\delta\sigma}$ \cite{KrupskiXXXXc}. \rosteree \STORY{../s/c0004/s0030/e0010} %%%%%%%%% %%%%%%%%% END %%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{ Compactifications of the real line and the real half-line} \begin{itemize} \item [(a)] Waraszkiewicz spirals \end{itemize} \section{ Other examples} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% %%%%%%%%% START of CHAPTER %%%%%%%%% /c0005 %%%%%%%%% %%%%%%%%% \chapter{ USC DECOMPOSITIONS OF CONTINUA} \section{ Decomposition spaces} \section{ Upper semi-continuous decompositions} \section{ Continuous decompositions} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% %%%%%%%%% START of CHAPTER %%%%%%%%% /c0006 %%%%%%%%% %%%%%%%%% \chapter{ CURVES} \section{ Order of a space at a point} \begin{itemize} \item [(a)] Curves of a finite order \item [(b)] Regular continua %Sierpi\'nski gasket = triangle curve; local dendrites \item [(c)] Rational continua \item [(d)] Suslinian continua \item [(e)] Urysohn examples $X(n)$, $X(\aleph_0)$; $X(2) = \Bbb S^1$ \end{itemize} \section{ Case-Chamberlin examples} \section{ Curves without arcs} \section{ Covering defined classes (inverse limit expansions)} \begin{itemize} \item [(a)] Arc-like continua \item [(b)] Circle-like continua (the circle, the Warsaw circle and other elementary examples; the circle of pseudo-arcs) \item [(c)] Tree-like continua (dendrites, dendroids, $\lambda$-dendroids; Ingram's example, Bellamy's example) \end{itemize} \section{ Other examples} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% %%%%%%%%% START of CHAPTER %%%%%%%%% /c0007 %%%%%%%%% %%%%%%%%% \chapter{ FAMILIES OF CONTNINUA} \section{ Universal elements} \begin{itemize} \item [(a)] Universal arc-like continuum %(Schori) \item [(b)] Universal circle-like continuum \item [(c)] Universal hereditarily indecomposable continuum %(Ma\'ckowiak) \item [(d)] Universal $\Cal P$-compacta %(McCord) \item [(e)] Universal smooth dendroids \item [(f)] Universal smooth fan \item [(g)] Sierpi\'nski universal plane curve \item [(h)] Menger universal curve \end{itemize} \section{ Common models} \section{ Families of incomparable continua} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% %%%%%%%%% START of CHAPTER %%%%%%%%% /c0008 %%%%%%%%% %%%%%%%%% \chapter{ PLANE CONTINUA} \section{ Embeddability of continua in surfaces} \section{ Planability in general} \section{ Planability of graphs on of locally connected continua} \begin{itemize} \item [(a)] Kuratowski's graphs \item [(b)] Claytor's examples of nonplanable curves \end{itemize} \section{ Separation of the plane} \begin{itemize} \item [(a)] Common boundary of plane domains \item [(b)] Kuratowski's example \item [(c)] Hereditary cut of the plane (Whyburn's curve) \end{itemize} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% %%%%%%%%% START of CHAPTER %%%%%%%%% /c0009 %%%%%%%%% %%%%%%%%% \chapter{ MAPPINGS} \section{ Various kinds of mappings and relations between them} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% %%%%%%%%% START of CHAPTER %%%%%%%%% /c0010 %%%%%%%%% %%%%%%%%% \chapter{ CONTINUA WITH SPECIAL MAPPING PROPERTIES} \section{ Existence of special mappings} \begin{itemize} \item [(a)] Retractions from hyperspaces \item [(b)] Means \item [(c)] Contractibility (also hereditary) \item [(d)] Selectibility \end{itemize} \section{ Self-similar continua} \section{ Self-homeomorphic continua} \section{ Rigid continua} \begin{itemize} \item [(a)] Cook's continua \end{itemize} \section{ Chaotic continua} \section{ Absolute retracts; spaces $LC^n$ and $LC^{\infty}$} \section{ Absolute retracts for various classes of continua} \section{ Class (W) and other classes defined by mappings} \section{ Homogeneity} \section{ Variations of homogeneity } \section{ Generalized homogeneity} \section{ Embeddability in $\Bbb R^n$} \section{ Weakly chainable continua} \section{ Fixed point property} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% %%%%%%%%% START of CHAPTER %%%%%%%%% /c0011 %%%%%%%%% %%%%%%%%% \chapter{ CONTINUA AS FRACTALS} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% %%%%%%%%% START of CHAPTER %%%%%%%%% /c0012 %%%%%%%%% %%%%%%%%% \chapter{ DYNAMICAL SYSTEMS} \section{ Attractors} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% %%%%%%%%% START of CHAPTER %%%%%%%%% /c0013 %%%%%%%%% %%%%%%%%% \chapter{ OTHER PROPERTIES} \section{ Colocal connectedness} \section{ Aposyndesis and its variations} \section{ Property of Kelley (also hereditary)} \section{ Smoothness} \section{ Arc-smoothness} \section{ $C^*$-smoothness} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% %%%%%%%%% START of CHAPTER %%%%%%%%% /c0014 %%%%%%%%% %%%%%%%%% \chapter{ NEW FIGURES} \section{ New examples 1} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% START %%%%%%%%% %%%%%%%%% /c0014/s0010/e0010 %%%%%%%%% %%%%%%%%% \subsection{ {Whyburn's curve $W$} } \setcounter{equation}{0} \e{Whyburn's curve} \FIGURE[scale=1]{}{../s/c0014/s0010/e0010/}{A}{Whyburn's curve} \STORY{../s/c0014/s0010/e0010} %%%%%%%%% %%%%%%%%% END %%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% START %%%%%%%%% %%%%%%%%% /c0014/s0010/e0020 %%%%%%%%% %%%%%%%%% \subsection{ {Pseudo-arc} } \setcounter{equation}{0} \e{Pseudo-arc} \FIGURE[scale=1]{}{../s/c0014/s0010/e0020/}{A} {Pseudo-arc} \FIGURE[scale=1]{}{../s/c0014/s0010/e0020/}{B} {Pseudo-arc} \FIGURE[scale=1]{}{../s/c0014/s0010/e0020/}{C} {Pseudo-arc} \FIGURE[scale=1]{}{../s/c0014/s0010/e0020/}{D} {Pseudo-arc} \FIGURE[scale=1]{}{../s/c0014/s0010/e0020/}{E} {Pseudo-arc} \FIGURE[scale=1]{}{../s/c0014/s0010/e0020/}{F} {Pseudo-arc} \STORY{../s/c0014/s0010/e0020} %%%%%%%%% %%%%%%%%% END %%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% START %%%%%%%%% %%%%%%%%% /c0014/s0010/e0040 %%%%%%%%% %%%%%%%%% \subsection{ {Zig-zag circle} } \setcounter{equation}{0} \e{zig-zag circle} \FIGURE[scale=1]{}{../s/c0014/s0010/e0040/}{A}{zig-zag circle} \STORY{../s/c0014/s0010/e0040} %%%%%%%%% %%%%%%%%% END %%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% START %%%%%%%%% %%%%%%%%% /c0014/s0010/e0050 %%%%%%%%% %%%%%%%%% \subsection{ {M-continuum} } \setcounter{equation}{0} \e{M-continuum} \FIGURE[scale=1]{}{../s/c0014/s0010/e0050/}{A}{M-continuum} \STORY{../s/c0014/s0010/e0050} %%%%%%%%% %%%%%%%%% END %%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% START %%%%%%%%% %%%%%%%%% /c0014/s0010/e0070 %%%%%%%%% %%%%%%%%% \subsection{ {Cantor interaction} } \setcounter{equation}{0} \e{Cantor interaction} \FIGURE[scale=1]{}{../s/c0014/s0010/e0070/}{A}{Cantor interaction} \GIF[scale=1]{}{../s/c0014/s0010/e0070/}{AA}{Cantor interaction} \GIF[scale=1]{}{../s/c0014/s0010/e0070/}{AAA}{Cantor interaction} \STORY{../s/c0014/s0010/e0070} %%%%%%%%% %%%%%%%%% END %%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% START %%%%%%%%% %%%%%%%%% /c0014/s0010/e0080 %%%%%%%%% %%%%%%%%% \subsection{ {Pseudo-circle} } \setcounter{equation}{0} \e{pseudo-circle} \FIGURE[scale=1]{}{../s/c0014/s0010/e0080/}{A}{pseudo-circle} \STORY{../s/c0014/s0010/e0080} %%%%%%%%% %%%%%%%%% END %%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% START %%%%%%%%% %%%%%%%%% /c0014/s0010/e0090 %%%%%%%%% %%%%%%%%% \subsection{ {Semicircle continuum} } \setcounter{equation}{0} \e{semicircle continuum} \FIGURE[scale=1]{}{../s/c0014/s0010/e0090/}{A}{semicircle continuum} \STORY{../s/c0014/s0010/e0090} %%%%%%%%% %%%%%%%%% END %%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% START %%%%%%%%% %%%%%%%%% /c0014/s0010/e0100 %%%%%%%%% %%%%%%%%% \subsection{ {Boxes continuum} } \setcounter{equation}{0} \e{boxes continuum} \FIGURE[scale=1]{}{../s/c0014/s0010/e0100/}{A}{boxes continuum} \STORY{../s/c0014/s0010/e0100} %%%%%%%%% %%%%%%%%% END %%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{ New examples 2} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% START %%%%%%%%% %%%%%%%%% /c0014/s0020/e0010 %%%%%%%%% %%%%%%%%% \subsection{ {Graph $K_{3,3}$} } \setcounter{equation}{0} \es{Graph $K_{3,3}$}{graph K33} \FIGURE[scale=1]{}{../s/c0014/s0020/e0010/}{A}{Graph $K_{3,3}$} \GIF[scale=1]{}{../s/c0014/s0020/e0010/}{AA}{Graph $K_{3,3}$} \STORY{../s/c0014/s0020/e0010} %%%%%%%%% %%%%%%%%% END %%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% START %%%%%%%%% %%%%%%%%% /c0014/s0020/e0020 %%%%%%%%% %%%%%%%%% \subsection{ {Graph $K_5$} } \setcounter{equation}{0} \es{Graph $K_5$}{graph K5} \FIGURE[scale=1]{}{../s/c0014/s0020/e0020/}{A}{Graph $K_5$} \GIF[scale=1]{}{../s/c0014/s0020/e0020/}{AA}{Graph $K_5$} \STORY{../s/c0014/s0020/e0020} %%%%%%%%% %%%%%%%%% END %%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% START %%%%%%%%% %%%%%%%%% /c0014/s0020/e0030 %%%%%%%%% %%%%%%%%% \subsection{ {Space filling curve} } \setcounter{equation}{0} \e{space filling curve} \FIGURE[scale=1]{}{../s/c0014/s0020/e0030/}{A}{space filling curve} \STORY{../s/c0014/s0020/e0030} %%%%%%%%% %%%%%%%%% END %%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% START %%%%%%%%% %%%%%%%%% /c0014/s0020/e0040 %%%%%%%%% %%%%%%%%% \subsection{ {Universal arc-like continuum} } \setcounter{equation}{0} \e{universal arc-like continuum} \FIGURE[scale=1]{}{../s/c0014/s0020/e0040/}{A}{universal arc-like continuum} \STORY{../s/c0014/s0020/e0040} %%%%%%%%% %%%%%%%%% END %%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% START %%%%%%%%% %%%%%%%%% /c0014/s0020/e0050 %%%%%%%%% %%%%%%%%% \subsection{ {Lakes of Vada} } \setcounter{equation}{0} \e{lakes of Vada} \FIGURE[scale=1]{}{../s/c0014/s0020/e0050/}{A}{lakes of Vada} \STORY{../s/c0014/s0020/e0050} %%%%%%%%% %%%%%%%%% END %%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% START %%%%%%%%% %%%%%%%%% /c0014/s0020/e0060 %%%%%%%%% %%%%%%%%% \subsection{ {Borsuk fan} } \setcounter{equation}{0} \e{Borsul fan} \FIGURE[scale=1]{}{../s/c0014/s0020/e0060/}{A}{Borsuk fan} \GIF[scale=1]{}{../s/c0014/s0020/e0060/}{AA}{Borsuk fan} \STORY{../s/c0014/s0020/e0060} %%%%%%%%% %%%%%%%%% END %%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% START %%%%%%%%% %%%%%%%%% /c0014/s0020/e0070 %%%%%%%%% %%%%%%%%% \subsection{ {Ingram's atriodic continuum} } \setcounter{equation}{0} \e{Ingram's atriodic continuum} \FIGURE[scale=1]{}{../s/c0014/s0020/e0070/}{A}{Ingram's atriodic continuum} \GIF[scale=1]{}{../s/c0014/s0020/e0070/}{AA}{Ingram's atriodic continuum} \STORY{../s/c0014/s0020/e0070} %%%%%%%%% %%%%%%%%% END %%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% START %%%%%%%%% %%%%%%%%% /c0014/s0020/e0080 %%%%%%%%% %%%%%%%%% \subsection{ {Double Warsaw circle} } \setcounter{equation}{0} \e{double Warsaw circle} \FIGURE[scale=1]{}{../s/c0014/s0020/e0080/}{A}{double Warsaw circle} \STORY{../s/c0014/s0020/e0080} %%%%%%%%% %%%%%%%%% END %%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% START %%%%%%%%% %%%%%%%%% /c0014/s0020/e0090 %%%%%%%%% %%%%%%%%% \subsection{ {Warsaw circle} } \setcounter{equation}{0} \e{Warsaw circle} \FIGURE[scale=1]{}{../s/c0014/s0020/e0090/}{A}{Warsaw circle} \STORY{../s/c0014/s0020/e0090} %%%%%%%%% %%%%%%%%% END %%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% START %%%%%%%%% %%%%%%%%% /c0014/s0020/e0100 %%%%%%%%% %%%%%%%%% \subsection{ {Crooked Warsaw circle} } \setcounter{equation}{0} \e{crooked Warsaw circle} \FIGURE[scale=1]{}{../s/c0014/s0020/e0100/}{A}{crooked Warsaw circle} \STORY{../s/c0014/s0020/e0100} %%%%%%%%% %%%%%%%%% END %%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{ New examples 3} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% START %%%%%%%%% %%%%%%%%% /c0014/s0030/e0010 %%%%%%%%% %%%%%%%%% \subsection{ {Young spiral} } \setcounter{equation}{0} \e{Young spiral} \FIGURE[scale=1]{}{../s/c0014/s0030/e0010/}{A}{Young spiral} \STORY{../s/c0014/s0030/e0010} %%%%%%%%% %%%%%%%%% END %%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% START %%%%%%%%% %%%%%%%%% /c0014/s0030/e0020 %%%%%%%%% %%%%%%%%% \subsection{ {Young space} } \setcounter{equation}{0} \e{Young space} \FIGURE[scale=1]{}{../s/c0014/s0030/e0020/}{A}{Young space} \STORY{../s/c0014/s0030/e0020} %%%%%%%%% %%%%%%%%% END %%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% START %%%%%%%%% %%%%%%%%% /c0014/s0030/e0030 %%%%%%%%% %%%%%%%%% \subsection{ {Bing space} } \setcounter{equation}{0} \e{Bing space} \FIGURE[scale=1]{}{../s/c0014/s0030/e0030/}{A}{Bing space} \STORY{../s/c0014/s0030/e0030} %%%%%%%%% %%%%%%%%% END %%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% START %%%%%%%%% %%%%%%%%% /c0014/s0030/e0040 %%%%%%%%% %%%%%%%%% \subsection{ {Universal smooth dendroid} } \setcounter{equation}{0} \e{universal smooth dendroid} \FIGURE[scale=1]{}{../s/c0014/s0030/e0040/}{A}{universal smooth dendroid} \STORY{../s/c0014/s0030/e0040} %%%%%%%%% %%%%%%%%% END %%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% START %%%%%%%%% %%%%%%%%% /c0014/s0030/e0050 %%%%%%%%% %%%%%%%%% \subsection{ {Claytor's curve A} } \setcounter{equation}{0} \e{Claytor's curve A} \FIGURE[scale=1]{}{../s/c0014/s0030/e0050/}{A}{Claytor's curve A} \STORY{../s/c0014/s0030/e0050} %%%%%%%%% %%%%%%%%% END %%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% START %%%%%%%%% %%%%%%%%% /c0014/s0030/e0060 %%%%%%%%% %%%%%%%%% \subsection{ {Claytor's curve B} } \setcounter{equation}{0} \e{Claytor's curve B} \FIGURE[scale=1]{}{../s/c0014/s0030/e0060/}{A}{Claytor's curve B} \STORY{../s/c0014/s0030/e0060} %%%%%%%%% %%%%%%%%% END %%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% START %%%%%%%%% %%%%%%%%% /c0014/s0030/e0070 %%%%%%%%% %%%%%%%%% \subsection{ {Sin addiction continuum} } \setcounter{equation}{0} \e{sin addiction continuum} \FIGURE[scale=1]{}{../s/c0014/s0030/e0070/}{A}{sin addiction continuum} \STORY{../s/c0014/s0030/e0070} %%%%%%%%% %%%%%%%%% END %%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% START %%%%%%%%% %%%%%%%%% /c0014/s0030/e0080 %%%%%%%%% %%%%%%%%% \subsection{ {Dumbbell} } \setcounter{equation}{0} \e{dumbbell} \FIGURE[scale=1]{}{../s/c0014/s0030/e0080/}{A}{dumbbell} \STORY{../s/c0014/s0030/e0080} %%%%%%%%% %%%%%%%%% END %%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% START %%%%%%%%% %%%%%%%%% /c0014/s0030/e0090 %%%%%%%%% %%%%%%%%% \subsection{ {Figure Eight} } \setcounter{equation}{0} \e{figure eight} \FIGURE[scale=1]{}{../s/c0014/s0030/e0090/}{A}{figure eight} \STORY{../s/c0014/s0030/e0090} %%%%%%%%% %%%%%%%%% END %%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% START %%%%%%%%% %%%%%%%%% /c0014/s0030/e0100 %%%%%%%%% %%%%%%%%% \subsection{ {Noose} } \setcounter{equation}{0} \e{noose} \FIGURE[scale=1]{}{../s/c0014/s0030/e0100/}{A}{noose} \STORY{../s/c0014/s0030/e0100} %%%%%%%%% %%%%%%%%% END %%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{ New examples 4} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% START %%%%%%%%% %%%%%%%%% /c0014/s0040/e0010 %%%%%%%%% %%%%%%%%% \subsection{ {Theta curve} } \setcounter{equation}{0} \e{theta curve} \FIGURE[scale=1]{}{../s/c0014/s0040/e0010/}{A}{theta curve} \STORY{../s/c0014/s0040/e0010} %%%%%%%%% %%%%%%%%% END %%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% START %%%%%%%%% %%%%%%%%% /c0014/s0040/e0020 %%%%%%%%% %%%%%%%%% \subsection{ {Trapezoid basins continuum} } \setcounter{equation}{0} \e{trapezoid basins continuum} \FIGURE[scale=1]{}{../s/c0014/s0040/e0020/}{A}{trapezoid basins continuum} \STORY{../s/c0014/s0040/e0020} %%%%%%%%% %%%%%%%%% END %%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% START %%%%%%%%% %%%%%%%%% /c0014/s0040/e0030 %%%%%%%%% %%%%%%%%% \subsection{ {Dragon continuum} } \setcounter{equation}{0} \e{dragon continuum} \FIGURE[scale=1]{}{../s/c0014/s0040/e0030/}{A}{dragon continuum} \FIGURE[scale=1]{}{../s/c0014/s0040/e0030/}{B}{dragon continuum} \GIF[scale=1]{}{../s/c0014/s0040/e0030/}{BB}{dragon continuum} L. Fearnley, D. G. Wright : Geometric realization of a Bellamy Continuum, BUll. London Math. Soc. 15(1993), 177-183. \STORY{../s/c0014/s0040/e0030} %%%%%%%%% %%%%%%%%% END %%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{ New examples 5} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% START %%%%%%%%% %%%%%%%%% /c0014/s0050/e0010 %%%%%%%%% %%%%%%%%% \subsection{ {Harmonic fan} } \setcounter{equation}{0} \e{harmonic fan} \FIGURE[scale=1]{}{../s/c0014/s0050/e0010/}{A}{harmonic fan} \STORY{../s/c0014/s0050/e0010} %%%%%%%%% %%%%%%%%% END %%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% START %%%%%%%%% %%%%%%%%% /c0014/s0050/e0020 %%%%%%%%% %%%%%%%%% \subsection{ {Harmonic shredded fan} } \setcounter{equation}{0} \e{harmonic shredded fan} \FIGURE[scale=1]{}{../s/c0014/s0050/e0020/}{A}{harmonic shredded fan} \STORY{../s/c0014/s0050/e0020} %%%%%%%%% %%%%%%%%% END %%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% START %%%%%%%%% %%%%%%%%% /c0014/s0050/e0030 %%%%%%%%% %%%%%%%%% \subsection{ {Cantor fan} } \setcounter{equation}{0} \e{Cantor fan} \FIGURE[scale=1]{}{../s/c0014/s0050/e0030/}{A}{Cantor fan} \STORY{../s/c0014/s0050/e0030} %%%%%%%%% %%%%%%%%% END %%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% START %%%%%%%%% %%%%%%%%% /c0014/s0050/e0040 %%%%%%%%% %%%%%%%%% \subsection{ {Lelek fan} } \setcounter{equation}{0} \e{Lelek fan} \FIGURE[scale=1]{}{../s/c0014/s0050/e0040/}{A}{Lelek fan} \GIF[scale=1]{}{../s/c0014/s0050/e0040/}{AA}{Lelek fan} \STORY{../s/c0014/s0050/e0040} %%%%%%%%% %%%%%%%%% END %%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{ New examples 6} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% START %%%%%%%%% %%%%%%%%% /c0014/s0060/e0010 %%%%%%%%% %%%%%%%%% \subsection{ {Sierpi\'nski triangle} } \setcounter{equation}{0} \es{Sierpi\'nski triangle}{Sierpinski triangle} \FIGURE[scale=1]{}{../s/c0014/s0060/e0010/}{A}{Sierpi\'nski triangle} \GIF[scale=1]{}{../s/c0014/s0060/e0010/}{AA}{Sierpi\'nski triangle} \STORY{../s/c0014/s0060/e0010} %%%%%%%%% %%%%%%%%% END %%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% START %%%%%%%%% %%%%%%%%% /c0014/s0060/e0020 %%%%%%%%% %%%%%%%%% \subsection{ {Makuchowski umbrella} } \setcounter{equation}{0} \e{Makuchowski umbrella} \FIGURE[scale=1]{}{../s/c0014/s0060/e0020/}{A}{Makuchowski umbrella} \STORY{../s/c0014/s0060/e0020} %%%%%%%%% %%%%%%%%% END %%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% START %%%%%%%%% %%%%%%%%% /c0014/s0060/e0040 %%%%%%%%% %%%%%%%%% \subsection{ {Snake} } \setcounter{equation}{0} \e{snake} \FIGURE[scale=1]{}{../s/c0014/s0060/e0040/}{A}{snake} \STORY{../s/c0014/s0060/e0040} %%%%%%%%% %%%%%%%%% END %%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% START %%%%%%%%% %%%%%%%%% /c0014/s0060/e0060 %%%%%%%%% %%%%%%%%% \subsection{ {Cantor snake} } \setcounter{equation}{0} \e{Cantor snake} \FIGURE[scale=1]{}{../s/c0014/s0060/e0060/}{A}{Cantor snake} \STORY{../s/c0014/s0060/e0060} %%%%%%%%% %%%%%%%%% END %%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% START %%%%%%%%% %%%%%%%%% /c0014/s0060/e0080 %%%%%%%%% %%%%%%%%% \subsection{ {Cruller} } \setcounter{equation}{0} \e{cruller} \FIGURE[scale=1]{}{../s/c0014/s0060/e0080/}{A}{cruller} \STORY{../s/c0014/s0060/e0080} %%%%%%%%% %%%%%%%%% END %%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{ New examples 7} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% START %%%%%%%%% %%%%%%%%% /c0014/s0070/e0010 %%%%%%%%% %%%%%%%%% \subsection{ {Cantor comb} } \setcounter{equation}{0} \e{Cantor comb} \FIGURE[scale=1]{}{../s/c0014/s0070/e0010/}{A}{Cantor comb} \STORY{../s/c0014/s0070/e0010} %%%%%%%%% %%%%%%%%% END %%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% START %%%%%%%%% %%%%%%%%% /c0014/s0070/e0020 %%%%%%%%% %%%%%%%%% \subsection{ {Gehman dendrite on ski} } \setcounter{equation}{0} \e{Gehman dendrite on ski} \FIGURE[scale=1]{}{../s/c0014/s0070/e0020/}{A}{Gehman dendrite on ski} \STORY{../s/c0014/s0070/e0020} %%%%%%%%% %%%%%%%%% END %%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% START %%%%%%%%% %%%%%%%%% /c0014/s0070/e0030 %%%%%%%%% %%%%%%%%% \subsection{ {Makuchowski twin} } \setcounter{equation}{0} \e{Makuchowski twin} \FIGURE[scale=1]{}{../s/c0014/s0070/e0030/}{A}{Makuchowski twin} \STORY{../s/c0014/s0070/e0030} %%%%%%%%% %%%%%%%%% END %%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% START %%%%%%%%% %%%%%%%%% /c0014/s0070/e0040 %%%%%%%%% %%%%%%%%% \subsection{ {Cantor function comb} } \setcounter{equation}{0} \e{Cantor function comb} \FIGURE[scale=1]{}{../s/c0014/s0070/e0040/}{A}{Cantor function comb} \STORY{../s/c0014/s0070/e0040} %%%%%%%%% %%%%%%%%% END %%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% %%%%%%%%% START %%%%%%%%% %%%%%%%%% /c0014/s0070/e0050 %%%%%%%%% %%%%%%%%% \subsection{ {Oversteegen's dendroid} } \setcounter{equation}{0} \e{Oversteegen's dendroid} \FIGURE[scale=1]{}{../s/c0014/s0070/e0050/}{A}{Oversteegen's dendroid} \FIGURE[scale=1]{}{../s/c0014/s0070/e0050/}{B}{Oversteegen's dendroid - a bunch of Cantor function combs} \FIGURE[scale=1]{}{../s/c0014/s0070/e0050/}{C}{Oversteegen's dendroid - from above (observe that the Cantor function combs have teeths shifted using the $z^2$ function, starting from left to right)} \GIF[scale=1]{}{../s/c0014/s0070/e0050/}{CC}{Oversteegen's dendroid - an animation} Lex G. Oversteegen, \textit{Open retractions and locally confluent mappings of certain continua}, Houston J. Math. \textbf{6},1 (1980), 113--125. \STORY{../s/c0014/s0070/e0050} %%%%%%%%% %%%%%%%%% END %%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % START DEFINITIONS % \appendix \chapter{Definitions} %% %% \section*{ absolute end point } A point $p$ of a \g{continuum} $X$ is called an \h{absolute end point} of $X$ if $X \setminus \{p\}$ is a \g{composant} of $X$. \section*{ absolute retract } An \h{absolute retract} is a space $X$ such that if $X$ is embedded as a closed subset $X'$ of a space $Y$, then $X'$ is a \g{retract} of $Y$. \section*{ absolutely terminal continuum } A proper \g{subcontinuum} $K$ of a \g{continuum} $X$ is said to be an \h{absolutely terminal continuum} of $X$ provided that $K$ is a \g{terminal continuum} of each \g{subcontinuum} $L$ of $X$ which properly contains $K$. \section*{ acyclic } A \cm $X$ is said to be \h{acyclic} provided that each mapping from $X$ into the unit circle $\Bbb S^1$ is homotopic to a constant mapping, i.e., for all mappings $f: X \to \Bbb S^1$ and $c: X \to \{p\} \subset \Bbb S^1$ there exists a mapping $h: X \times [0,1] \to \Bbb S^1$ such that for each point $x \in X$ we have $h(x,0) = f(x)$ and $h(x,1) = c(x)$. \section*{ almost chainable } A \g{continuum} $M$ is \h{almost chainable} if, for each positive number $\varepsilon$, there exists an open cover $D$ of $M$ with mesh $\varepsilon$ and a chain $C$ of elements of $D$ and an end link $L$ of $C$ such that no element of $D \setminus C$ intersects any link of $C$ other than $L$ and every point of $M$ is at a distance less than $\varepsilon$ from some link of $C$. \section*{ aposyndetic } A connected space $X$ is \h{aposyndetic at} $H$ with respect to $K$ if there is a closed connected subset of $X$ with $H$ in its interior and not intersecting $K$, and $X$ is \h{aposyndetic} if it is aposyndetic at each point with respect to every other point. A \cm $X$ is said to be \emph{aposyndetic} provided that for each point $p \in X$ and for each $q \in X \setminus \{p\}$ there exists a subcontinuum $K$ of $X$ and an open set $U$ of $X$ such that $p \in U \subset K \subset X \setminus \{q\}$ (see e.g.\cite[Exercise 1.22, p. 12]{Nadler1992a}). \section*{ arc } An \h{arc} is any space which is homeomorphic to the closed interval $[0,1]$. \section*{ arc-smooth } Given a \cm $X$ with an \g{arc-structure} $A$, the pair $(X,A)$ (see \g{arc-structure}) is said to be \h{arc-smooth at a point} $v \in X$ provided that the induced function $A_v: X \to C(X)$ defined by $A_v (x) = A(v,x)$ is continuous. Then the point $v$ is called an \h{initial point} of $(X,A)$. The pair $(X,A)$ is said to be \h{arc-smooth} provided that there exists a point in $X$ at which $(X,A)$ is arc-smooth. An arbitrary space $X$ is said to be \emph{arc-smooth at a point} $v \in X$ provided that there exists an arc-structure $A$ on $X$ for which $(X,A)$ is arc-smooth at $v$. The space $X$ is said to be \emph{arc-smooth} if it is arc-smooth at some point (see \cite[p. 546]{Fugate+1981a}). Note that a dendroid is \g{smooth} if and only if it is arc-smooth. \section*{ arc-structure } By an \h{arc-structure} on an arbitrary space $X$ we understand a function $A : X \times X \to C(X)$ such that for every two distinct points $x$ and $y$ in $X$ the set $A(x,y)$ is an \g{arc} from $x$ to $y$ and that the following metric-like axioms are satisfied for every points $x$, $y$ and $z$ in $X$: \begin{itemize} \item [(1)] $A (x,x) = \{x\}$; \item [(2)] $A (x,y) = A (y,x)$; \item [(3)] $A (x,z) \subset A(x,y) \cup A (y,z)$, \newline with equality prevailing whenever $y \in A (x,z)$. \end{itemize} We put $(X,A)$ to denote that the space $X$ is equipped with an arc-structure $A$ (see \cite[p. 546]{Fugate+1981a}). Note that if there exists an arc-structure on a continuum, then the \cm is arcwise connected. \section*{ atomic } A surjective mapping $f: X \to Y$ between compact spaces is said to be (see \cite[Chapter 3 and 4, p. 12-28]{Mackowiak1979a}): \newline -- \h{atomic} provided that for each subcontinuum $K$ of $X$ such that the set $f(K)$ is nondegenerate we have $K = f^{-1}(f(K))$; \newline \section*{ atriodic } a \g{continuum} that contains no \g{subcontinuum} which is itself a \g{triod} is said to be \h{atriodic} \section*{ biconnected } A space is \h{biconnected} if it is not the sum of two mutually exclusive nondegenerate connected point sets. \section*{ bihomogeneous } Suppose $X$ is a space such that, for each two points $a$ and $b$, there is a homeomorphism of $X$ onto itself such that $h(a) = b$ and $h(b) = a$. Then $X$ is \h{bihomogeneous}. \section*{ Borel sets } The family $F$ of \h{Borel sets} of a space $S$ is the smallest family satisfying the conditions: (i) Every closed set belongs to $F$; (ii) If $X$ is in $F$, then $S \setminus X$ is in $F$; (iii) The countable intersection of elements of $F$ belongs to $F$. \section*{ branch point } A \h{branch point} of a \g{continuum} is the vertex of a \g{simple triod} lying in that \g{continuum}. \section*{ chainable } A \h{simple chain} is a finite sequence $L_1, L_2, \cdots,L_n$ of open sets such that $L_i$ intersects $L_j$ if and only if $|i-j| \leq 1$. The terms of the sequence $L_1, L_2, \cdots,L_n$ are called the \h{links} of the chain. An $\varepsilon$\h{-chain} is a chain each of whose links has diameter less than $\varepsilon$. a \g{continuum} $M$ is \h{chainable} if, for each positive number $\varepsilon$, $M$ can be covered by an $\varepsilon$-chain. \section*{ chaotic } A nondegenerate topological space $X$ is said to be \h{chaotic} if for any two distinct points $p$ and $q$ of $X$ there exist an open neighborhood $U$ of $p$ and an open neighborhood $V$ of $q$ such that no open subset of $U$ is homeomorphic to any open subset of $V$. \section*{ circle } A \h{circle} is any space which is homeomorphic to the unit circle. \section*{ completely regular } a \g{continuum} is \h{completely regular} if each nondegenerate \g{subcontinuum} has a nonempty interior. \section*{ composant } If a point $p$ of a \g{continuum} $X$ is given, then the \h{composant of $X$ belonging to} $p$ is defined to be the union of all proper \g{subcontinua} of $X$ which contain $p$. A set is a \h{composant of} $X$ provided it is the composant of $X$ belonging to $p$ for some point $p$ of $X$. \section*{ confluent } A mapping $f$ of a compact space $X$ onto a compact space $Y$ is \h{confluent} if, for each continuum $C$ in $Y$, each component of $f^{-1}(C)$ is mapped onto $C$ by $f$. \section*{ -connected } A topological space $(X,\tau)$ is said to be $\sigma$\h{-connected} provided that $X$ is not the union of more than one and at most countably many nonempty, mutually disjoint, closed subsets. A \g{continuum} is $\delta${\it-connected} if each two of its points can be connected by an \g{hereditarily decomposable} \g{irreducible} \g{subcontinuum}. \section*{ connected im kleinen } A space is \h{connected im kleinen} if it is aposyndetic at each point with respect to each closed point set not containing that point. \section*{ continuous selection of } Let a \cm $X$, a compact space $Y$ and a function $F: X \to 2^Y$ be given. A function $f: X \to Y$ is called a \h{continuous selection of} $F$ provided that it is \g{continuous} and $f(x) \in F(x)$ for each $x \in X$. \section*{ converges homeomorphically to the continuum } The statement that the sequence $M_1, M_2, \cdots$ \h{converges homeomorphically to the continuum} $M$ means there exists a sequence $h_1, h_2, \cdots$ of homeomorphisms such that, for each positive integer $i$, $h_i$ is a homeomorphism from $M_i$ onto $M$ and for each positive number $\varepsilon$ there exists a positive integer $N$ such that if $j > N$ then, for all $x$, dist$(h_j(x),x) < \varepsilon$. \section*{ converge $0$-regularly } A sequence of sets $\{A_n\}$ contained in a metric space $X$ with a metric $d$ is said to \h{converge $0$-regularly} to its limit $A = \Lim A_n$ (see \g{hyperspace} ) provided that for each $\eps > 0$ there is a $\delta > 0$ and there is an index $n_0 \in \N$ such that if $n > n_0$ then for every two points $p, q \in A_n$ with $d(p,q) < \delta$ there is a connected set $C_n \subset A_n$ satisfying conditions $p, q \in C_n$ and $\diam C_n < \eps$ (see \cite[Chapter 9, \S 3, p. 174]{Whyburn1942a}). \section*{ convex } Given a \cm $X$ with an \g{arc-structure} $A$, a subset $Z$ of $X$ is said to be \h{convex} provided that for each pair of points $x$ and $y$ of $Z$ the arc $A(x,y)$ is a subset of $Z$. If $Z$ is a convex subcontinuum of $X$, then $A|Z \times Z$ is an \g{arc-structure} on $Z$. We define $X$ to be \h{locally convex at a point} $p \in X$ provided that for each open set $U$ containing $p$ there is a \g{convex} set $Z$ such that $p \in \inter Z \subset \cl Z \subset U$ (see \cite[I.2, p. 548-549]{Fugate+1981a}). \section*{ continuum } A \h{continuum} is a nonempty, compact, connected metric space. \section*{ contractible } A \cm $X$ is said to be \h{contractible} provided there is a homotopy $h: X \times [0,1] \to X$ such that for some point $p \in X$ we have $h(x,0) = x$ and $h(x,1) = p$ for each $x \in X$ (see e.g. \cite[(16.2), p. 532]{Nadler1978a}). \section*{ curve } A \h{curve} is any \g{one-dimensional continuum}. \section*{ cut point } Let $S,T)$ be a connected topological space, and let $p \in S$. If $S \setminus \{p\}$ is connected, then $p$ is called a \h{non-cut point of} $S$. If $S \setminus \{p\}$ is not connected, then $p$ is called a \h{cut point of} $S$. \section*{ $C^*${-smooth} } Let $X$ be a \cmk Define $C^*: C(X) \to C(C(X))$ by $C^*(A) = C(A)$. It is known that for any \cm $X$ the function $C^*$ is upper semi-continuous, \cite[Theorem 15.2, p. 514]{Nadler1978a}, and it is continuous on a dense $G_\delta$ subset of $C(X)$, \cite[Corollary 15.3, p. 515]{Nadler1978a}. A \cm $X$ is said to be $C^*$\h{-smooth at} $A \in C(X)$ provided that the function $C^*$ is continuous at $A$. A \cm $X$ is said to be $C^*$\h{-smooth} provided that the function $C^*$ is continuous on $C(X)$, i.e., at each $A \in C(X)$ (see \cite[Definition 5.15, p. 517]{Nadler1978a}). \section*{ cyclic element } Let a \cm $X$ be locally connected and let $p \in X$. By a \h{cyclic element} of $X$ containing $p$ we mean either the singleton $\{p\}$ if $p$ is a cut point or an end point of $X$, or the set consisting of $p$ and of all points $x \in X$ such that no point of $X$ cuts $X$ between $p$ and $x$, otherwise (see e.g. \cite[Chapter 4, \S 2, p. 66]{Whyburn1942a}). \section*{ decomposable } a \g{continuum} $X$ is said to be \h{decomposable} provided that $X$ can be written as the union of two proper \g{subcontinua}. \section*{ deformation retract of } Let $X$ be a space and let $A$ and $B$ be subspaces of $X$ with $A \subset B$. Then $A$ is called a \h{deformation retract of} $B$ over $X$ provided that the identity mapping $i_B: B \to B$ is homotopic in $X$ to a \g{retract}ion $r: B \to A$. Further, $A$ is called a \h{strong deformation retract of} $B$ over $X$ provided that it is a deformation \g{retract} of $B$ over $X$ and the homotopy keeps the points of $A$ fixed throughout the entire deformation of $B$ into $A$ (see e.g. \cite[Definition 6.3, p. 324]{Dugundji1966a}). \section*{ dendroid } A \h{dendroid} is an \g{arcwise connected}, \g{hereditarily unicoherent} \g{continuum}. \section*{ dendrite } A \h{dendrite} means a locally connected \g{continuum} containing no \g{simple closed curve}. \section*{ dispersion point } A point $p$ of a nondegenerate connected space $X$ is an \h{explosion point}, or \h{dispersion point}, of $X$ provided that $X \setminus \{p\}$ is totally disconnected. \section*{ end point } %A point $p$ of a \g{continuum} $X$ is called an %\h{end point} of $X$ if for each two \g{subcontinua} of $X$ both %containing $p$, one of the \g{subcontinua} contains the other. Point of \g{order} 1 in a continuum $X$ is called \hi{end point} {point!end} of $X$; the set of all end points of $X$ is denoted by $E(X)$. \section*{ equivalent } Let $\mathfrak M$ be a class of mappings. Two spaces $X$ and $Y$ are said to be \emph{equivalent with respect to} $\mathfrak M$ (shortly $\mathfrak M$-\h{equivalent}) if there are two mappings, both in $\mathfrak M$, one from $X$ onto $Y$ and the other from $Y$ onto $X$. If $\mathfrak M$ means the class of monotone mappings, we say that $X$ and $Y$ are \hi{monotonely equivalent} {equivalent!monotonely}. \section*{ $F$-equivalent } If $X$ and $Y$ are \g{continua}, we say that $Y$ is $F$\h{-equivalent} to $X$ provided there is a mapping in $F$ from $X$ onto $Y$ and a mapping in $F$ from $Y$ onto $X$. \section*{ feebly monotone } Let $X$ and $Y$ be continua. A mapping $f: X \to Y$ is said to be \hi{feebly monotone} {monotone!feebly} provided that if $A$ and $B$ are proper subcontinua of $Y$ such that $Y = A \cup B$, then their inverse images $f^{-1}(A)$ and $f^{-1}(B)$ are connected. \section*{ finitely Suslinean } a \g{continuum} $X$ is \h{finitely Suslinean} if, for each positive number $\varepsilon$, $X$ does not contain infinitely many mutually exclusive \g{subcontinua} of diameter greater than $\varepsilon$. \section*{ finitely linear } A function $f\: I \to Y$, where $I$ is a closed interval of the real line, is called \h{finitely linear} provided there exists a positive integer $m$ such that $I$ can be decomposed, for each $\varepsilon > 0$, into a finite number of closed subintervals $I_1, I_2, \cdots, I_k$ each of length less than $\varepsilon$ and with the property that the set $f(I_i)$ meets at most $m$ of the sets $f(I_1), f(I_2), \cdots, f(I_k)$ for $i = 1, 2, \cdots, k$. \section*{ fixed point } We say that a function $F: X \to 2^X$ (or a function $F: X \to C(X)$) \emph{has a} \h{fixed point} provided that there is a point $x \in X$ such that $x \in F(x)$. \par \section*{ fixed set property } A topological space $X$ is said to have the \h{fixed set property} for a certain class $C$ of maps of $X$ onto itself provided there exists, for each non-empty closed set $A$ in $X$, a map $f$ in $C$ such that $f(x) = x$ if and only if $x$ is in $A$. \section*{ fixed point self-homeomorphic } A topological space $X$ is called \hi{fixed point self-homeomorphic} {self-homeomorphic!fixed!point} if for any point $p$, any neighborhood $U$ of $p$, there is an embedding $h : X \to U$ with $h(p)=p$ and $p \in \hbox{\rm int} \, h(X)$. \section*{ fixed ball self-homeomorphic } A topological space $X$ is called \hi{fixed ball self-homeomorphic} {self-homeomorphic!fixed!ball} if for any point $p$, any neighborhood $U$ of $p$, there is a neighborhood $V$ of $p$ with $V \subseteq U$ and an embedding $h : X \to U$ satisfying $h|_V=\hbox{\rm id}\, _V$. \section*{ free arc } A \h{free arc} in $X$ is any subset of $X$ of the form $A \setminus \{x,y\}$ where $A$ is an arc in $X$ with \g{end points} $x$ and $y$ and $A \setminus \{x,y\}$ is open in $X$. \section*{ graph } A \h{graph} is a \g{continuum} which can be written as the union of finely many \g{arc}s any two of which are either disjoint or intersect only in one or both their \g{end point}s. \section*{ half-ray curve } a \g{continuum} is a \h{half-ray curve} if it is a continuous 1-1 image of the nonnegative reals. \section*{ hereditary } A property of a \cm $X$ is said to be \h{hereditary} if every subcontinuum of $X$ has the property. In particular, a \cm is said to be \emph{hereditarily unicoherent} if the intersection of any two of its subcontinua is connected. \section*{ hereditarily } Let $\mathfrak M$ be a class of mappings between continua. A mapping $f: X \to Y$ between continua is said to be \h{hereditarily} $\mathfrak M$ provided that its restriction to any subcontinuum of the domain $X$ is in $\mathfrak M$ (see \cite[Chapter 4, Section B, p. 16]{Mackowiak1979a}). \section*{ hereditarily decomposable } a \g{continuum} $X$ is said to be \h{hereditarily decomposable} provided that each nondegenerate \g{subcontinuum} of $X$ (as well as $X$ itself) is \g{decomposable}. \section*{ hereditarily equivalent } a \g{continuum} is \h{hereditarily equivalent} if it is homeomorphic to each of its nondegenerate \g{subcontinua}. \section*{ hereditarily indecomposable } a \g{continuum} $X$ is said to be \h{hereditarily indecomposable} provided that each of its \g{subcontinua} is \g{indecomposable},\ that is, for each subcontinuum $C \subset X$ and for every continua $A$ and $B$ such that $A \cup B = C$ we have either $A = C$ or $B = C$. \section*{ hereditarily locally connected } A \g{continuum} $X$ is said to be \h{hereditarily locally connected}, written \h{hlc}, provided that every \g{subcontinuum} of $X$ is a \g{Peano Continuum}. \section*{ hereditarily unicoherent } A connected topological space is said to be \h{hereditarily unicoherent} provided that each of its closed, connected subsets is \g{unicoherent}. \section*{ homogeneous } Let $\mathfrak M$ be a class of mappings. A space $X$ is said to be \hi{homogeneous with respect to} {homogeneous!with respect to} $\mathfrak M$ (or shortly $\mathfrak M$-\emph{homogeneous}) provided that for every two points $p$ and $q$ of $X$ there is a surjective mapping $f: X \to X$ such that $f(p) = q$ and $f \in \mathfrak M$. If $\mathfrak M$ is the class of homeomorphisms, we get the concept of a \h{homogeneous} space. \section*{ HU-terminal } A \g{subcontinuum} $K$ of a \g{hereditarily unicoherent} \g{continuum} $X$ is said to be a \h{HU-terminal continuum} of $X$ provided that $K$ is contained in an \g{irreducible} \g{subcontinuum} of $X$ and for every \g{irreducible} \g{subcontinuum} $I$ of $X$ containing $K$ there is a point $x\in X$ such that $I$ is \g{irreducible about} the union $K\cup \{x\}$. \section*{ hyperspace } Given a \cm $X$ with a metric $d$, we let $2^X$ to denote the \h{hyperspace} of all nonempty closed subsets of $X$ equipped with the \h{Hausdorff metric} $H$ defined by $$ H(A,B) = \max \{\sup \{d(a,B): a \in A \},\, \sup \{d(b,A): b \in B\}\} \quad \hbox {for} \; A, B \in 2^X $$ (see e.g. \cite[(0.1), p. 1 and (0.12), p. 10]{Nadler1978a}). If $H(A, A_n)$ tends to zero as $n$ tends to infinity, we put $A = \Lim A_n$. Further, we denote by $F_1(X)$ the hyperspace of singletons of $X$, and by $C(X)$ the hyper\-space of all subcontinua of $X$, i.e., of all connected elements of $2^X$. Since $X$ is homeomorphic to $F_1(X)$, there is a natural embedding of $X$ into $C(X)$, and so we can write $X \subset C(X) \subset 2^X$. Thus one can consider a \g{retract}ion from either $C(X)$ or $2^X$ onto $X$. \section*{ indecomposable } a \g{continuum} $X$ is said to be \h{indecomposable} provided that $X$ cannot be written as the union of two proper \g{subcontinua}. \section*{ induced mapping } Let $f: X \to Y$ be a mapping between continua. Then the \h{induced mapping} $C(f): C(X) \to C(Y)$ is defined by $C(f)(A) = f(A)$, where $A$ in the left member of the equality means an element of $C(X)$, while in the right one it is understood as a subcontinuum of $X$ (see \cite[(0.49), p. 23]{Nadler1978a}). \section*{ inverse limit } The reader can find necessary information on the \g{inverse limit}s of inverse sequences e.g. in Section 2 of the second chapter of \cite{Nadler1992a}, as well as in \cite{Kuratowski1968a}. \section*{ irreducible } Let $X$ be a \g{continuum} and $A\subset X$. Then, $X$ is said to be \h{irreducible about} $A$ provided no proper \g{subcontinuum} of $X$ contains $A$. a \g{continuum} $X$ is said to be \h{irreducible} provided that $X$ is \g{irreducible about} $\{p,q\}$ for some $p,q \in X$. \section*{ light } A mapping is \h{light} if each point inverse is totally disconnected. \section*{ like } If $X$ is a metric space, a mapping $f$ from $X$ to a space $Y$ is an $\varepsilon$\h{-map} if, for each point $y$ of $Y$, $\diam(f^{-1}(y)) \leq \varepsilon$. If $C$ is a collection of continua, a \g{continuum} $M$ is $C$\h{-like} if, for every positive number $\varepsilon$, there exists an $\varepsilon$-map of $M$ onto an element of $C$. In particular, a \g{continuum} is \h{tree-like} if, for some collection $C$ of trees, $M$ is $C${-like}. A concept of a tree-like \cm can be defined in several (equivalent) ways. One of them is the following. A \cm $X$ is said to be \emph{tree-like} provided that for each $\eps > 0$ there is a tree $T$ and a surjective mapping $f: X \to T$ such that $f$ is an $\eps$-mapping (i.e., $\diam f^{-1}(y) < \eps$ for each $y \in T$). Let us mention that a \cm $X$ is tree-like if and only if it is the \g{inverse limit} of an inverse sequence of \g{tree}s with surjective \g{bonding mapping}s. Compare e.g. \cite[p. 24]{Nadler1992a}. Using a concept of a nerve of a covering, one can reformulate the above definition saying that a \cm $X$ is be tree-like provided that for each $\eps > 0$ there is an $\eps$-covering of $X$ whose nerve is a \g{tree}. \par Finally, the original definition using tree-chains can be found e.g. in Bing's paper \cite[p. 653]{Bing1951a}. \section*{ locally } Let $\mathfrak M$ be a class of mappings between compact spaces. A surjective mapping $f: X \to Y$ between continua is said to be \h{locally} $\mathfrak M$ provided that for each point $x \in X$ there is a closed \nbh $V$ of $x$ such that $f(V)$ is is a closed \nbh of $f(x)$ and that the restriction $f|V$ is in $\mathfrak M$ (see \cite[Chapter 4, Section C, p. 18]{Mackowiak1979a}). \section*{ locally confluent } A mapping $f$ from a space $X$ onto a space $Y$ is \h{locally confluent} if, for each point $y$ of $Y$, there is an open set $U$ containing $y$ such that $f^{-1}(\, \Cl{U} \,)$ is \g{confluent}. \section*{ local homeomorphism } A surjective mapping $f: X \to Y$ between compact spaces is said to be (see \cite[Chapter 3 and 4, p. 12-28]{Mackowiak1979a}): \newline -- a \h{local homeomorphism} provided that for each point $x \in X$ there exists a an open \nbh $U$ of $x$ such that $f(U)$ is an open \nbh of $f(x)$ and that $f$ restricted to $U$ is a homeomorphism between $U$ and $f(U)$. \section*{ local separating point } A point $p$ of a locally compact separable metric space $L$ is a \h{local separating point} of $L$ provided there exists an open set $U$ of $L$ containing $p$ and two points $x$ and $y$ of the component containing $p$ of $U$ such that $U \setminus \{ p \}$ is the sum of two mutually separated point sets, one containing $x$ and the other containing $y$. \section*{ MO-mapping } A surjective mapping $f: X \to Y$ between compact spaces is said to be (see \cite[Chapter 3 and 4, p. 12-28]{Mackowiak1979a}): \newline -- an \h{MO-mapping} provided that it can be represented as the composition of two mappings, $f = f_2 \circ f_1$ such that $f_1$ is open and $f_2$ is monotone. \section*{ monotone map} A continuous function $f:X \to Y$ is said to be \hs{monotone}{monotone map} provided $f^{-1}(y)$ is connected for each $y \in Y$. \section*{ mutually aposyndetic } a \g{continuum} $M$ is \h{mutually aposyndetic} if, for each two points $A$ and $B$ of $M$, there exist mutually exclusive \g{subcontinua} $H$ and $K$ of $M$ containing $A$ and $B$, respectively, in their interiors. a \g{continuum} is \h{strictly non-mutually aposyndetic} if each two of its \g{subcontinua} with interiors intersect. \section*{ natural projection } Let continua $X$ and $Y$ be given. A mapping $f: X \times Y \to X$ is called the \h{natural projection} provided that it is defined by $f((x,y)) = x$. \section*{ neat } A \g{dendroid} $X$ is said to be \h{neat} provided that each one of its subdendroids has no \g{improper shore point} (see \cite[p. 939]{Neumann+1993a}). Let $\mathfrak M$ be a class of mappings. A class $\mathfrak M$ of mappings is said to be \emph{neat} provided that if all homeomorphisms are in $\mathfrak M$ and the composition of any two mappings in $\mathfrak M$ is also in $\mathfrak M$. \section*{ near homeomorphism } A mapping $f: X \to Y$ is called a \hi{near homeomorphism} {homeomorphism!near} provided that for each $\eps > 0$ there is a homeomorphism $h: X \to Y$ such that $\sup \{d(f(x), h(x)): x \in X\} < \eps$. \section*{ OM-mapping } A surjective mapping $f: X \to Y$ between compact spaces is said to be (see \cite[Chapter 3 and 4, p. 12-28]{Mackowiak1979a}): \newline -- an \h{OM-mapping} provided that it can be represented as the composition of two mappings, $f = f_2 \circ f_1$ such that $f_1$ is monotone and $f_2$ is open. \section*{ open map } A mapping $f$ of a space $X$ into a space $Y$ is \hs{open}{open map} if, for each open $A$ in $X$ the image $f(A)$ is open in $Y$. \section*{ openly minimal } We say that a \g{dendrite} $X$ is \hi{openly minimal} {minimal!openly} provided that every open image of $X$ can be openly mapped onto $X$. \section*{ orbit } Given a space $X$ let $\mathcal H(X)$ stand for the group of autohomeomorphisms of $X$. If a point $p \in X$ is fixed, then $\{h(p) \in X: h \in \mathcal H(X)\}$ is called an \h{orbit of} $p$. Orbits of points of $X$ either are mutually disjoint or coincide, and their union is the whole $X$. \section*{ order} Let $(X,T)$ be a topological space, and let $A \subset X$. Let $\beta$ be a cardinal number. We say that $A$ \h{is of order less than equal to} $\beta$ in $X$, written ord($A,X$)$\le \beta$, provided that for each $U \in T$ such that $A\subset U$, there exists $V\in T$ such that $A \subset V \subset U$ and $|Bd(V)|\le \beta$. We say that $A$ {\it is of order $\beta$ in $X$}, written ord($A,X$)$= \beta$, provided that ord($A,X$)$\le \beta$ and ord($A,X$)$\not \le \alpha$ for any cardinal number $\alpha < \beta$. A concept of an \h{order} of a point $p$ in a \cm $X$ (in the sense of Menger-Urysohn), written $\ord (p,X)$, is defined as follows. Let $\mathfrak n$ stand for a cardinal number. We write: \par $\ord (p,X) \le \mathfrak n$ provided that for every $\eps > 0$ there is an open \nbh $U$ of $p$ such that $\diam U \le \eps$ and $\card \bd U \le \mathfrak n$; \par $\ord (p,X) = \mathfrak n$ provided that $\ord (p,X) \le \mathfrak n$ and for each cardinal number $\mathfrak m < \mathfrak n$ the condition $\ord (p,X) \le \mathfrak m$ does not hold; \par $\ord (p,X) = \omega$ provided that the point $p$ has arbitrarily small open \nbhs $U$ with finite boundaries $\bd U$ and $\card \bd U$ is not bounded by any $n \in \N$. \par Thus, for any \cm $X$ we have $$ \ord (p,X) \in \{1, 2, \dots , n, \dots , \omega, \aleph_0, 2^{\aleph_0}\} $$ (convention: $\omega < \aleph_0$); see \cite[\S 51, I, p. 274]{Kuratowski1968a}. Let a \g{dendroid} $X$ and a point $p \in X$ be given. Then $p$ is said to be a \emph{point of order at least $\mathfrak m$ in the classical sense} provided that $p$ is the center of an $\mathfrak m$\g{-od} contained in $X$. We say that $p$ is a \emph{point of order $\mathfrak m$ in the classical sense} provided that $\mathfrak m$ is the minimum cardinality for which the above condition is satisfied (see \cite[p. 229]{Charatonik1962a}). \section*{ order preserving mapping } For each point $p$ of a \cm $X$ equipped with an \g{arc-structure} $A: X \times X \to C(X)$ we define a partial order $\le_p$ by letting $x \le_p y$ whenever $A(p,x) \subset A(p,y)$. Let $X$ and $Y$ be continua with fixed arc-structures $A$ and $B$, respectively. We say that a surjective mapping $f: X \to Y$ is a $\le_p$-\emph{mapping} provided that $x \le_p y$ in $X$ implies that $f(x) \le_{f(p)} f(y)$ in $Y$. If, in addition, $Y \subset X$, $B = A|(Y \times Y)$, and $f$ is a retraction, then $f$ is called a $\le_p$-\emph{retraction} (or $\le_p$-\h{preserving retraction}). The concept of a $<_p$-\emph{mapping} is defined in a similar manner (with $f(x) \ne f(y)$ implied by $x \ne y$). For \h{order preserving mapping}s see e.g. \cite[I.7, p. 553]{Fugate+1981a}. \section*{ ordinary point } Point of order 2 is called \hi{ordinary point} {point!ordinary} of $X$; the set of all ordinary points of $X$ is denoted by $O(X)$. \section*{ periodic point of } Let a \cm $X$ and a mapping $f: X \to X$ be given. For each natural number $n$ denote by $f^n$ the $n$-th iteration of $f$. A point $p \in X$ is called a \hi{periodic point} {point!periodic} \emph{of} $f$ provided that there is $n \in \N$ such that $f^n(p) = p$ The set of periodic points of a mapping $f: X \to X$ are denoted by $P(f)$. \section*{ periodic-recurrent property } The set of \g{periodic point}s and of \g{recurrent point}s of a mapping $f: X \to X$ are denoted by $P(f)$ and $R(f)$ respectively. Clearly $P(f) \subset R(f)$. A \cm $X$ is said to have the \hi{periodic-recurrent property} {property!periodic-recurrent} (shortly \h{PR-property}) provided that for every mapping $f: X \to X$ the equality $\cl P(f) = \cl R(f)$ holds. \section*{ pointwise self-homeomorphic } A topological space $X$ is called \hi{pointwise self-homeomorphic at a point} {self-homeomorphic!pointwise!at a point} $x \in X$ if for any neighborhood $U$ of $x$ there is a set $V$ such that $x\in V \subseteq U$ and $V$ is homeomorphic to $X$. The space $X$ is called \hi{pointwise self-homeomorphic} {self-homeomorphic!pointwise} if it is pointwise self-homeomorphic at each of its points. \section*{ pseudo-arc } A \h{pseudo-arc} is the only (up to homeomorphism) \g{hereditarily indecomposable} \g{arc-like} \g{continuum}. \section*{ pseudo-confluent } A mapping from $X$ to $Y$ is \h{pseudo-confluent} if every \g{irreducible} \g{continuum} in $Y$ is the image of a \g{continuum} in $X$. \section*{ property of Kelley } A \cm $X$ is said to have the \h{property of Kelley} provided that for each point $x \in X$, for each subcontinuum $K$ of $X$ containing $x$ and for each sequence of points $x_n$ converging to $x$ there exists a sequence of subcontinua $K_n$ of $X$ containing $x_n$ and converging to the continuum $K$ (see e.g. \cite[Definition 16.10, p. 538]{Nadler1978a}). \section*{ quasi-monotone } A surjective mapping $f: X \to Y$ between compact spaces is said to be (see \cite[Chapter 3 and 4, p. 12-28]{Mackowiak1979a}) \h{quasi-monotone} provided that for each subcontinuum $Q$ of $Y$ with the nonempty interior the set $f^{-1}(Q)$ has a finite number of components and $f$ maps each of them onto $Q$. \section*{ radially convex at } Given a \cm $X$ with an \g{arc-structure} $A$ and a point $p \in X$, a metric $d$ on $X$ is said to be \h{radially convex at} $p$ provided that $d(p,z) = d(p,y) + d(y,z)$ for every points $y, z \in X$ with $y \in A(p,z)$ (see \cite[I.4, p. 551]{Fugate+1981a}). \section*{ ramification point } Point of order at least 3 is called \hi{ramification point} {point!ramification} of $X$; the set of all ramification points of $X$ is denoted by $R(X)$. \section*{ rational } A \g{continuum} is \h{rational} if every two of its points can be separated by a countable point set. \section*{ real curve } A \g{continuum} is a \h{real curve} if it is a continuous 1-1 image of the real line. \section*{ recurrent point of } Let a \cm $X$ and a mapping $f: X \to X$ be given. For each natural number $n$ denote by $f^n$ the $n$-th iteration of $f$. A point $p \in X$ is called a \hi{recurrent point} {point!recurent} \emph{of} $f$ provided that for every \nbh $U$ of $p$ there is $n \in \N$ such that $f^n(p) \in U$. The set of recurrent points of a mapping $f: X \to X$ are denoted by $R(f)$. \section*{ refinable (monotonely)} A map $r$ from a compact metric space $X$ onto a compact metric space $Y$ is {\it(monotonely) }\h{refinable} if, for each positive number $\varepsilon$, there is a (\g{monotone}) $\varepsilon$-map $f$ from $X$ onto $Y$ such that, for each $x$ in $X$, $$ d\( f\(x\),r\(x\) \) < \varepsilon . $$ \section*{ regular } If $X$ is a \g{continuum} and $p\in X$, then $X$ is said to be \h{regular at} $p$ provided that there is a local base $\Cal L_p$ at $p$ such that the boundary of each member of $\Cal L_p$ is of finite cardinality. A continuum is said to be \h{regular} provided that $X$ is regular at each of its points. A \g{continuum} is \g{regular} if each two of its points can be separated by a finite point set. \section*{ retract, retraction } Let $X$ and $Y$ be continua. A mapping $f: X \to Y$ is said to be a \h{retraction} provided that $Y \subset X$ and the restriction $f|Y: Y \to f(Y) \subset X$ is the identity; then $Y$ is called a \h{retract} of $X$. \section*{ rigid } A nondegenerate topological space $X$ is said to be \h{rigid} if it has a trivial autohomeomorphism group, i.e., if the only homeomorphism of $X$ onto $X$ is the identity. \section*{ selectible } A \cm $X$ is said to be \h{selectible} provided that there exists a mapping $\sigma: C(X) \to X$ (called a \h{selection} for $C(X)$) such that $\sigma (A) \in A$ for each continuum $A \subset X$ (see e.g. \cite[p. 253]{Nadler1978a}). \section*{ scattered } A point set $X$ is \h{scattered} if every subset $Y$ of $X$ has a point that is not a limit point of $Y$. \section*{ self-homeomorphic } A topological space $X$ is called \h{self-homeomorphic} if for any open set $U \subseteq X$ there is a set $V \subseteq U$ such that $V$ is homeomorphic to $X$. \section*{ semi-aposyndetic } A space $X$ is \h{semi-aposyndetic} if, for each two of its points, $X$ is \g{aposyndetic} at one of them with respect to the other. \section*{ semi-confluent } A surjective mapping $f: X \to Y$ between compact spaces is said to be (see \cite[Chapter 3 and 4, p. 12-28]{Mackowiak1979a}): \newline -- \h{semi-confluent} provided that for each subcontinuum $Q$ of $Y$ and for every two components $C_1$ and $C_2$ of $f^{-1}(Q)$ either $f(C_1) \subset f(C_2)$ or $f(C_2) \subset f(C_1)$. \section*{ semi-continuum } A \h{semi-continuum} is a \g{continuum}wise connected point set. \section*{ semi-continuous } Let a \cm $X$, a compact space $Y$ and a function $F: X \to 2^Y$ be given. Put $$ F^{-1}(B) = \{x \in X: F(x) \cap B \ne \emptyset\}. $$ The function $F$ is said to be \emph{lower (upper)} \h{semi-continuous} provided that $F^{-1}(B)$ is open (closed) for each open (closed) subset $B \subset Y$. It is said to be \h{continuous} provided that it is both lower and upper semi-continuous. This notion of continuity agrees with the one for mappings between metric spaces. \section*{ semi-locally connected } A \g{continuum} $M$ is \h{semi-locally connected} at the point $p$ if, for every domain $D$ containing $p$, there exits a domain $E$ lying in $D$ and containing $p$ such that $M \setminus E$ has only a finite number of components. A continuum is \h{totally non-semi-locally connected} if it is not \g{semi-locally connected} at any point. \section*{ simple closed curve } A \h{simple closed curve} is any space which is homeomorphic to the unit circle. \section*{ simple $\mathfrak m$-od } Let $\mathfrak m$ be a cardinal number. By a \emph{simple} $\mathfrak m$\h{-od} with the center $p$ we mean the union of $\mathfrak m$ arcs every two of which have $p$ as the only common point. \section*{ simple triod } The union of three arcs emanating from a point v is called a \h{simple triod} provided the singleton {v} is the intersection of any two of the arcs ; the point v is then called the \h{top} of the simple triod, and the arcs are called its \h{arms}. \section*{ shore point } A point $p$ of a \g{dendroid} $X$ is called a \h{shore point} of $X$ if there exists a sequence of subdendroids $X_n$ of $X \setminus \{p\}$ such that $X = \Lim X_n$. A shore point of $X$ that is not an \g{end point} of $X$ is called an \h{improper shore point} of $X$ (see \cite[p. 939]{Neumann+1993a}). \section*{ smooth (dendroid) } A dendroid $X$ is said to be \h{smooth at a point} $p \in X$ provided that for each point $x \in X$ and for each sequence of points $\{x_n\}$ in $X$ tending to $x$, the sequence of \g{arc}s $px_n$ tends to the arc $px$. A \g{dendroid} $X$ is said to be \h{smooth} provided that it is smooth at some point $p \in X$ (see \cite[p. 298]{Charatonik+1970a}). \section*{ strongly chaotic } A nondegenerate topological space $X$ is said to be \hi{strongly chaotic} {chaotic!strongly} if for any two distinct points $p$ and $q$ of $X$ there exist open neighborhoods $U$ of $p$ and $V$ of $q$ respectively such that no open subset of $U$ is homeomorphic to any subset of $V$; \section*{ strongly pointwise self-homeomorphic } A topological space $X$ is called \hi{strongly pointwise self-homeomorphic at a point} {self-homeomorphic!pointwise!strongly!at a point} $x \in X$ if for any neighborhood $U$ of $x$ there is a neighborhood $V$ of $x$ such that $x\in V \subseteq U$ and $V$ is homeomorphic to $X$. The space $X$ is called \hi{strongly pointwise self-homeomorphic} {self-homeomorphic!pointwise!strongly} if it is strongly pointwise self-homeomorphic at each of its points. \section*{ strongly rigid } A nondegenerate topological space $X$ is said to be \hi{strongly rigid} {rigid!strongly} if the only homeomorphism of $X$ into $X$ is the identity of $X$ onto itself. \section*{ strongly self-homeomorphic } A topological space $X$ is called \hi{strongly self-homeomorphic} {self-homeomorphic!strongly} if for any open set $U \subseteq X$ there is a set $V \subseteq U$ with nonempty interior such that $V$ is homeomorphic to $X$. \section*{ strongly unicoherent1 } A \g{continuum} $X$ is \h{strongly unicoherent} provided $X$ is \g{unicoherent} and each proper \g{subcontinuum} with interior is \g{unicoherent}. \section*{ strongly unicoherent2 } A \cm $X$ is said to be \h{strongly unicoherent2} provided that it is \g{unicoherent} and for each pair of its proper subcontinua $A$ and $B$ such that $X = A \cup B$, each of $A$ and $B$ is \g{unicoherent} (see \cite[p. 587]{Benett1971a}). \section*{ Suslinean } A \g{continuum} is \h{Suslinean} if it does not contain uncountably many mutually exclusive nondegenerate \g{subcontinua}. \section*{ $T_A (x)$ } For each point $x$ of a \cm $X$ with an \g{arc-structure} $A$ we define (see \cite[I.3, p. 550]{Fugate+1981a}) $T_A (x) = \{y \in X: $ each convex subcontinuum of $X$ with $y$ in its interior contains $\; x\}$. Since $T_A (x)$ is always closed, we have $T_A: X \to 2^X$. \par \section*{ terminal continuum } A proper \g{subcontinuum} $K$ of a \g{continuum} $X$ is said to be a \h{terminal continuum} of $X$ provided that if whenever $A$ and $B$ are proper \g{subcontinua} of $X$ having union equal to $X$ such that $A\cap K \ne \emptyset \ne B \cap K$, then either $X=A\cup K$ or $X=B\cup K$. \section*{ thin } A topological space $S$ is \h{thin} if and only if for each two homeomorphic subset $A$ and $B$ of $S$, there is a homeomorphism $h$ of $S$ onto itself such that $h(A)=B$. \section*{ translation } If $P$ and $Q$ are subspaces of a metric space $X$ with a metric $d$, and $\eps$ is a positive number, then a mapping $g: P \to Q$ is called an $\eps$-\h{translation} provided that $d(p, g(p)) < \eps$ for each point $p \in P$. \section*{ tree } A \h{tree} is a graph which contains no \g{simple closed curve}. A tree as a \g{one-dimensional} \g{acyclic} connected polyhedron, i.e., a \g{dendrite} with finitely many \g{end point}s. \section*{ triod } A \g{continuum} is called a \h{triod} provided it contains a \g{subcontinuum} whose complement is the union of three nonempty pairwise disjoint open sets. \section*{ unicoherent } A connected topological space $S$ is said to be \h{unicoherent} provided that whenever $A$ and $B$ are closed, connected subsets of $S$ such that $S=A\cup B$, then $A\cap B$ is connected. Let a \cm $X$ and its subcontinuum $Y$ be given. Then $X$ is said to be \hi{unicoherent at} {unicoherent!at} $Y$ provided that for each pair of proper subcontinua $A$ and $B$ of $X$ such that $X = A \cup B$ the intersection $A \cap B \cap Y$ is connected (see \cite[p. 146]{Owens1986a}). \section*{ uniformly continuum-chainable } A \g{continuum} $X$ is said to be \h{uniformly continuum-chainable} if for each positive number $\varepsilon$ there is an integer $k=k(\varepsilon)$ such that for each pair $x$,$y$ of points of $X$, there are subcontinua $A_1, \cdots, A_k$ of $X$ each of diameter less than $\varepsilon$ such that $x\in A_1$, $y\in A_k$ and $A_i \cap A_j \ne \emptyset$ whenewer $|i-j|\leq 1$. \section*{ uniquely arcwise connected } A \cm is said to be \h{uniquely arcwise connected} provided that for every two of its points there is exactly one arc in the \cm joining these points. \section*{ universal } Let a class $\mathcal S$ of spaces be given. A member $U$ of $\mathcal S$ is said to be \hi{universal for} {universal!for} $\mathcal S$ if every member of $\mathcal S$ can be embedded in $U$, i.e., if for every $X \in \mathcal S$ there exists a homeomorphism $h: X \to h(X) \subset U$. Accordingly, a dendrite is said to be \h{universal} if it contains a homeomorphic image of any other dendrite. \section*{ weak cut point } A point $x$ of a \g{continuum} $M$ is a \hi{weak cut point} {point!weak cut} of $M$ if there are two points $p$ and $q$ of $M$ such that every \g{subcontinuum} of $M$ that contains both $p$ and $q$ also contains $x$. \section*{ weakly chainable } A \g{continuum} is \h{weakly chainable} if it is a continuous image of a \g{chainable} continuum. \section*{ weakly confluent } A surjective mapping $f: X \to Y$ between compact spaces is said to be (see \cite[Chapter 3 and 4, p. 12-28]{Mackowiak1979a}) \h{weakly confluent} provided that for each subcontinuum $Q$ of $Y$ there is a component the set $f^{-1}(Q)$ which is mapped under $f$ onto $Q$. \section*{ weakly hereditarily unicoherent } A continuum is said to be \h{weakly hereditarily unicoherent} if the intersection of any two of its subcontinua with nonempty interiors is connected (see \cite[p. 152]{Owens1986a} and references therein). \section*{ weakly monotone } A surjective mapping $f: X \to Y$ between compact spaces is said to be (see \cite[Chapter 3 and 4, p. 12-28]{Mackowiak1979a}) \h{weakly monotone} provided that for each subcontinuum $Q$ of $Y$ with the nonempty interior each component the set $f^{-1}(Q)$ is mapped under $f$ onto $Q$. \section*{ weakly smooth at a point } A dendroid $X$ is said to be \h{weakly smooth at a point} $p \in X$ provided that the subspace of $2^X$ consisting of all subarcs of $X$ of the form $px$ for $x \in X$ is compact (see \cite[p. 113]{Lum1975a}). \section*{ weakly Suslinian } A \g{continuum} $X$ is said to be \h{weakly Suslinian} provided that $X$ is not the union of more than one pairwise disjoint nondegenerate subcontinua of $X$. \section*{ widely connected } A nondegenerate connected space $X$ is \h{widely connected} if each nondegenerate connected subset of $X$ is dense in $X$. \section*{ width $w(X)$ } For any compact metric space $X$, the \h{width $w(X)$} of $X$ is the l.u.b. of the set of all real numbers $a$ which satisfy the following condition: for each $\varepsilon > 0$, there exists a finite open cover $C$ of $X$ such that $\mesh(C) < \varepsilon$ and for each chain $C'$ which is a subcollection of $C$ there is a member $A$ of $C$ such that $d(A,{C'}^*) \geq a$. %%%%%%%%% END DEFINITIONS %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% START LIST OF EXAMPLES \chapter{List of examples} \esLIST{Graph $K_{3,3}$}{graph K33} \eLIST{}{Bing space} \eLIST{}{Borsul fan} \eLIST{}{boxes continuum} \eLIST{}{buckethandle} \eLIST{}{Cantor accordion} \eLIST{}{Cantor comb} \eLIST{}{Cantor fan} \eLIST{}{Cantor function comb} \eLIST{}{Cantor interaction} \eLIST{}{Cantor organ} \eLIST{}{Cantor snake} \eLIST{}{Claytor's curve A} \eLIST{}{Claytor's curve B} \esLIST{comb $W$}{comb W} \esLIST{comb $W_R$}{comb WR} \eLIST{}{crooked Warsaw circle} \eLIST{}{cruller} \esLIST{dendrite $D_m$}{dendrite Dm} \esLIST{dendrite $D_S$}{dendrite DS} \esLIST{dendrite $L_0$}{dendrite L0} \eLIST{}{dendrite of de Groot-Wille type} \esLIST{dendrite $P$}{dendrite P} \eLIST{}{double buckethandle} \eLIST{}{double Warsaw circle} \eLIST{}{dragon continuum} \eLIST{}{dumbbell} \eLIST{}{dyadic solenoid} \eLIST{}{figure eight} \eLIST{}{Gehman dendrite} \eLIST{}{Gehman dendrite on ski} \esLIST{Graph $K_5$}{graph K5} \eLIST{}{harmonic fan} \eLIST{}{harmonic shredded fan} \eLIST{}{horseshoe} \eLIST{}{Ingram's atriodic continuum} \eLIST{}{Knaster continuum} \eLIST{}{lakes of Vada} \eLIST{}{Lelek fan} \esLIST{locally connected combs}{locally connected comb} \eLIST{}{locally connected fan} \eLIST{}{Makuchowski twin} \eLIST{}{Makuchowski umbrella} \eLIST{}{M-continuum} \eLIST{}{Menger universal curve} \esLIST{Miller dendrite $S$}{Miller dendrite S} \eLIST{}{modified Miller dendrite} \eLIST{}{noose} \esLIST{dendrite $G_{\omega}$}{dendrite Gomega} \eLIST{}{Omiljanowski dendrite} \eLIST{}{Oversteegen's dendroid} \esLIST{$\mathbf p$-adic Knaster continuum}{p-adic Knaster continuum} \esLIST{$\mathbf p$-adic solenoid}{p-adic solenoid} \eLIST{}{Pseudo-arc} \eLIST{}{pseudo-circle} \eLIST{}{semicircle continuum} \esLIST{Sierpi\'nski carpet}{Sierpinski carpet} \esLIST{Sierpi\'nski triangle}{Sierpinski triangle} \esLIST{Sierpi\'nski universal plane curve}{Sierpinski universal plane curve} \eLIST{}{sin addiction continuum} \eLIST{}{sin curve} \eLIST{}{snake} \eLIST{}{solenoid} \eLIST{}{space filling curve} \esLIST{standard universal dendrite of order $m$}{standard universal dendrite of order m} \esLIST{standard universal dendrite of orders in $S$}{standard universal dendrite of orders in S} \eLIST{}{theta curve} \eSLIST{(topologist's)}{topologist's sin curve} \eLIST{}{trapezoid basins continuum} \eLIST{}{universal arc-like continuum} \eLIST{}{universal smooth dendroid} \eLIST{}{Warsaw circle} \esLIST{Wa\.zewski universal dendrite}{Wazewski universal dendrite} \eLIST{}{Whyburn's curve} \eLIST{}{Young space} \eLIST{}{Young spiral} \eLIST{}{zig-zag circle} %%%%%%%%% END LIST OF EXAMPLES %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{htmlonly} \chapter{Technical information} \begin{rawhtml}
Here you can find some useful Technical information concerning file structure, compiling, latex2html, tex style, definitions file, references file etc.
\end{rawhtml} \end{htmlonly} %% %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %% \printindex % % Alphabetization and navigation within the index % ...these special index entries must come *after* the \printindex % else half of the hyperlinks will point to the preceding page. % \begin{htmlonly} \newcommand{\indexAlpha}[5]{\index{#1@\htmlref{_}{#2}% \htmlref{\HTML{SUB}{\LARGE #3}}{AZ}\htmlref{_}{#4}\label{#5}| }} % \indexAlpha{\$}{Z}{\$}{dot}{doll}% \indexAlpha{.}{doll}{~.~}{A}{dot}% \indexAlpha{A}{dot}{A}{B}{A}% \indexAlpha{B}{A}{B}{C}{B}% \indexAlpha{C}{B}{C}{D}{C}% \indexAlpha{D}{C}{D}{E}{D}% \indexAlpha{E}{D}{E}{F}{E}% \indexAlpha{F}{E}{F}{G}{F}% \indexAlpha{G}{F}{G}{H}{G}% \indexAlpha{H}{G}{H}{I}{H}% \indexAlpha{I}{H}{I}{J}{I}% %\indexAlpha{I}{H}{I}{L}{I}% %\indexAlpha{J}{I}{J, K}{L}{K}% \indexAlpha{J}{I}{J}{K}{J}% \indexAlpha{K}{J}{K}{L}{K}% \indexAlpha{L}{K}{L}{M}{L}% \indexAlpha{M}{L}{M}{N}{M}% \indexAlpha{N}{M}{N}{O}{N}% \indexAlpha{O}{N}{O}{P}{O}% \indexAlpha{P}{O}{P}{Q}{P}% %\indexAlpha{P}{O}{P}{R}{P}% %\indexAlpha{Q}{P}{Q}{S}{R}% \indexAlpha{Q}{P}{Q}{R}{Q}% \indexAlpha{R}{Q}{R}{S}{R}% \indexAlpha{S}{R}{S}{T}{S}% \indexAlpha{T}{S}{T}{U}{T}% \indexAlpha{U}{T}{U}{V}{U}% \indexAlpha{V}{U}{V}{W}{V}% \indexAlpha{W}{V}{W}{X}{W}% \indexAlpha{X}{W}{X}{Y}{X}% %\indexAlpha{X}{W}{X}{Z}{X}% %\indexAlpha{Y}{X}{Y, Z}{doll}{Z}% \indexAlpha{Y}{X}{Y}{Z}{Y}% \indexAlpha{Z}{Y}{Z}{doll}{Z}% % % %% This is an alphabetical navigation panel. \index{@\label{AZ}\textbf{\LARGE \htmlref{\$}{doll} \htmlref{.}{dot} \htmlref{A}{A}% \htmlref{B}{B} \htmlref{C}{C} \htmlref{D}{D} \htmlref{E}{E} \htmlref{F}{F} % \htmlref{G}{G} \htmlref{H}{H} \htmlref{I}{I} \htmlref{J}{J} \htmlref{K}{K} % \htmlref{L}{L} \htmlref{M}{M} \htmlref{N}{N} \htmlref{O}{O} \htmlref{P}{P} % \htmlref{Q}{Q} \htmlref{R}{R} \htmlref{S}{S} \htmlref{T}{T} \htmlref{U}{U} % \htmlref{V}{V} \htmlref{W}{W} \htmlref{X}{X} \htmlref{Y}{Y} \htmlref{Z}{Z}}\\ \htmlrule[all]| } \end{htmlonly} %% %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% START REFERENCES \begin{thebibliography}{99} \bibliographystyle{alpha} %% %% \newcommand{\citej}{\emph} \newcommand{\citeb}{\rm} \newcommand{\citen}{\textbf} %%%%%%%%%%%% % % NEW REFERENCES !!!!!!!! % % PLEASE PUT NEW REFERENCES in any form HERE: (I will put them to the correct place and form) %%%%%%%%%%%% % % REFERENCES USED IN EXAMPLES !!!!!!!! % % PLEASE USE THIS REFERENCES IN THE FORM INDICATED IN the file RULES.TEX : \bibitem[Aarts et al. \- 1991]{Aarts+1991a} J. M. Aarts and R. J. Fokkink, \textit{On composants of the bucket handle}, Fund. Math. \textbf{139} (1991), 193--208. \bibitem[Aarts et al. \- 1998]{Aarts+1998a} J. M. Aarts and R. J. Fokkink, \textit{Fixed points of the bucket handle}, Proc. Amer. Math. Soc. \textbf{126} (1998), 881--885. \bibitem[Anderson \- 1957]{Anderson1957a} R. D. Anderson, \textit{Zero-dimensional compact groups of homeomorphisms}, Pacific J. Math. \textbf{7} (1957), 797--810. \bibitem[Anderson \- 1958]{Anderson1958a} R. D. Anderson, \textit{One-dimensional continuous curves and a homogeneity theorem}, Ann. of Math. \textbf{68} (1958), 1--16. \bibitem[Ar\'evalo et al. \- 2001]{Arevalo+2001a} D. Ar\'evalo, W. J. Charatonik, P. Pellicer Covarrubias and L. Sim\'on, \textit{Dendrites with a closed set of end points}, Topology Appl. \textbf{115}(2001), 1--17. \bibitem[Bandt \- 1994]{Bandt1994a} C. Bandt, \textit{Composants of the horseshoe}, Fund. Math. \textbf{144} (1994), 231--241. \bibitem[Barge \- 1986]{Barge1986a} M. Barge, \textit{Horseshoe maps and inverse limits}, Pacific J. Math. \textbf{156} (1986), 29--39. \bibitem[Barge \- 1988]{Barge1988a} M. Barge, \textit{A method for constructing attractors}, Ergodic Theory Dynam. Systems \textbf{8} (1988), 331--349. \bibitem[Bellamy \- 1973]{Bellamy1973a} D. P. Bellamy, \textit{Composants of Hausdorff indecomposable continua; a mapping approach}, Pacific J. Math. \textbf{47} (1973), 303--309. \bibitem[Bennett \- 1971]{Benett1971a} D. E. Bennett, \citej{Aposyndetic properties of unicoherent continua}, Pacific J. Math. \citen{37} (1971), 585--589. \bibitem[Bestvina \- 1988]{Bestvina1988a} M. Bestvina, \textit{Characterizing $k$-dimensional universal Menger compacta}, Mem. Amer. Math. Soc. \textbf{380} (1988). \bibitem[Bing \- 1951]{Bing1951a} R. H. Bing, \textit{Snake-like continua}, Duke Math. J. \textbf{18} (1951), 653--663. \bibitem[Borsuk \- 1959]{Borsuk1959a} K. Borsuk, \textit{Concerning the classification of topological spaces from the stand-point of the theory of retracts}, Fund. Math. \textbf{46} (1959), 321--330. \bibitem[Borsuk \- 1967]{Borsuk1967a} K. Borsuk, \textit{Theory of retracts}, Monografie Matematyczne \textbf{44}, PWN-Polish Scientific Publishers, Warszawa, 1967. \bibitem[Bothe \- 1963]{Bothe1963a} H. G. Bothe, \textit{Eine Einbettung $m$-dimensionalen Mengen in einen $(m+1)$-dimensionalen Absoluten Retrakt}, Fund. Math. \textbf{52} (1963), 209--224. \bibitem[Brechner \- 1966]{Brechner1966a} B. L. Brechner, \textit{On the dimension of certain spaces of homeomorphisms}, Trans. Amer. Math. Soc. \textbf{121} (1966), 515--548. \bibitem[Cannon \- 1973]{Cannon1973a} J. W. Cannon, \textit{A positional characterization of the $n-1$-dimensional Sierpi\'nski curve in $S^n$ ($n\ne 4$)}, Fund. Math. \textbf{79} (1973), 107--112. \bibitem[Charatonik \- 1962]{Charatonik1962a} J. J. Charatonik, \textit{On ramification points in the classical sense}, Fund. Math. \textbf {51} (1962), 229--252. \bibitem[Caratonik \- 1977]{Charatonik1977a} J. J. Charatonik, \textit{ A chaotic dendrite}, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. \textbf{25} (1977), 1261--1263. \bibitem[Charatonik \- 1979]{Charatonik1979a} J. J. Charatonik, \textit{On chaotic curves}, Colloq. Math. \textbf{41} (1979), 219--236. \bibitem[Chaaratonik \- 1980]{Charatonik1980a} J. J. Charatonik, \textit{Open mappings of universal dendrites}, Bull. Acad. Polon. Sci. Ser. Sci. Math. \textbf{28} (1980), 489--494. \bibitem[Charatonik \- 1984]{Charatonik1984a} J. J. Charatonik, \textit{Mappings of the Sierpi\'nski curve onto itself}, Proc. Amer. Math. Soc. \textbf{92} (1984), 125--132. \bibitem[Charatonik \- 1991]{Charatonik1991a} J. J. Charatonik, \textit{Monotone mappings of universal dendrites}, Topology Appl. \textbf{38} (1991), 163--187. \bibitem[Charatonik \- 1995]{Charatonik1995a} J. J. Charatonik, \textit{ Homeomorphisms of universal dendrites}, Rend. Circ. Mat. Palermo (2) \textbf{44} (1995), 457--468. \bibitem[Charatonik \- 1996]{Charatonik1996a} J. J. Charatonik, \textit{ An uncountable family of monotone homogeneous dendrites}, Glasnik Mat. \textbf{31 (51)} (1996), 289--294. \bibitem[Charatonik \- 1998]{Charatonik1998a} J. J. Charatonik, \textit{ On sets of periodic and of recurrent points}, Publ. Inst. Math. (Beograd) (N. S.) \textbf{63 (77)} (1998), 131--142. \bibitem[Charatonik \- 1999]{Charatonik1999a} J. J. Charatonik, \textit{ On chaotic dendrites}, Period. Math. Hungar. \textbf{38} (1999), 19--29. \bibitem[Charatonik \- 2000]{Charatonik2000a} J. J. Charatonik, \textit{ On generalized rigidity}, Houston J. Math. \textbf{26} (2000), 639--660. \bibitem[Charatonik et al. \- 1970]{Charatonik+1970a} J. J. Charatonik and C. Eberhart, \citej{On smooth dendroids}, Fund. Math. \citen{67} (1970), 297--322. \bibitem[Charatonik et al. \- 1990]{Charatonik+1990a} J. J. Charatonik, W. J. Charatonik and S. Miklos, \textit{Confluent mappings of fans}, Dissertationes Math. (Rozprawy Mat.) \textbf{301} (1990), 1--86. \bibitem[Charatonik et al. \- 1994]{Charatonik+1994a} J. J. Charatonik, W. J. Charatonik and J. R. Prajs, \textit{Mapping hierarchy for dendrites}, Dissertationes Math. (Rozprawy Mat.) \textbf{333} (1994), 1--52. \bibitem[Charatonik et al. \- 1996]{Charatonik+1996a} J. J. Charatonik and W. J. Charatonik, \textit{Strongly chaotic dendrites}, Colloq. Math. \textbf{70} (1996), 181--190. \bibitem[Charatonik et al. \- 1997a]{Charatonik+1997a} J. J. Charatonik and W. J. Charatonik, \textit{Monotone homogeneity of dendrites}, Comment. Math. Univ. Carolin. \textbf{38} (1997), 361--370. \bibitem[Charatonik et al. \- 1997b]{Charatonik+1997b} J. J. Charatonik and W. J. Charatonik, \textit{Periodic-recurrent property of some continua}, Bull. Austral. Math. Soc. \textbf{56} (1997), 109--118. \bibitem[Charatonik et al. \- 1998]{Charatonik+1998a} J. J. Charatonik and W. J. Charatonik, \textit{Dendrites}, Aportaciones Mat. Comun. \textbf{22} (1998), 227--253. \bibitem[Charatonik et al. \- 2000]{Charatonik+2000a} J. J. Charatonik, W. J. Charatonik and P. Krupski, \textit{Dendrites and light open mappings}, Proc. Amer. Math. Soc. \textbf{128} (2000), 1839--1843. \bibitem[Charatonik et al. XXXXa]{Charatonik+XXXXa} J. J. Charatonik, W. J. Charatonik and J. R. Prajs, \textit{Hereditarily unicoherent continua and their absolute retracts}, preprint. \bibitem[Charatonik et al. XXXXb]{Charatonik+XXXXb} J. J. Charatonik and P. Krupski, \textit{ Dendrites and light mappings}, preprint. \bibitem[Charatonik et al. XXXXc]{Charatonik+XXXXc} J. J. Charatonik and P. Krupski, \textit{On self-homeomorphic dendrites}, preprint. \bibitem[Charatonik W.J. et al. \- 1994]{CharatonikW+1994a} W. J. Charatonik and A. Dilks, \textit{On self-homeomorphic spaces}, Topology Appl. \textbf{55} (1994), 215--238. \bibitem[Chigogidze \- 1996]{Chigogidze1996a} A. Chigogidze, \textit{Inverse spectra}, North Holland, Elsevier, 1996. \bibitem[Chigogidze et al. \- 1995]{Chigogidze+1995a} A. Chigogidze, K. Kawamura and E. D. Tymchatyn, \textit{Menger manifolds}, Continua with the Houston problem book (H. Cook et al., eds.), Dekker, 1995, 37--88. \bibitem[Cook \- 1967]{Cook1967a} H. Cook, \textit{Upper semi-continuous continuum-valued mappings onto circle-like continua}, Fund. Math. \textbf{60} (1967), 233--239. \bibitem[D. van Dantzig \- 1930]{Dantzig1930a} D. van Dantzig, \textit{\"Uber topologisch homogene Kontinua}, Fund. Math. \textbf{15} (1930), 102--125. \bibitem[D\c ebski \- 1985]{Debski1985a} W. D\c ebski, \textit{On topological types of the simplest indecomposable continua}, Colloq. Math. \textbf{49} (1985), 203--211. \bibitem[D\c ebski \- 1992]{Debski1992a} W. D\c ebski, \textit{Two-to-one maps on solenoids and Knaster continua}, Fund. Math. \textbf{141} (1992), 277--285. \bibitem[Dilks et al. \- 1981]{Dilks+1981a} A. M. Dilks and J. T. Rogers, Jr., \textit{Whitney stability and contractible hyperspaces}, Proc. Amer. Math. Soc. \textbf{83} (1981), 633--640. \bibitem[Duda et al. \- 1991]{Duda+1991a} E. Duda, P. Krupski and J. T. Rogers, Jr., \textit{On locally chainable homogeneous continua}, Topology Appl. \textbf{42} (1991), 95--99 \bibitem[Dugundji \- 1966]{Dugundji1966a} J. Dugundji, \citeb{Topology}, Allyn and Bacon, Boston, 1966. \bibitem[Eberhart et al. \- 1999]{Eberhart+1999a} C. Eberhart, J. B. Fugate and S. Schumann, \textit{Open maps between Knaster continua}, Fund. Math. \textbf{162} (1999), 119--148. \bibitem[Emeryk \- 1980]{Emeryk1980a} A. Emeryk, \textit{Partitions of indecomposable continua into composants}, Proceedings of International Conference in Geometric Topology (Warsaw, 1978), PWN-Polish Scientific Publishers, Warszawa, 1980, 137--140. \bibitem[Engelking \- 1978]{Engelking1978a} R. Engelking, \textit{Dimension theory}, North Holland, Amsterdam and PWN-Polish Scientific Publishers, Warszawa, 1978. \bibitem[Fugate et al. \- 1981]{Fugate+1981a} J. B. Fugate, G. R. Gordh, Jr. and L. Lum, \citej{Arc-smooth continua}, Trans. Amer. Math. Soc. \citen{265} (1981), 545--561. \bibitem[Gehman \- 1925]{Gehman1925a} H. M. Gehman, \textit{Concerning the subsets of a plane continuous curve}, Ann. of Math. \textbf{27} (1925), 29--46. \bibitem[Godlewski \- 1970]{Godlewski1970a} S. Godlewski, \textit{Solenoids of comparable shapes are homeomorphic}, Bull. Acad. Polon. Sci. S\'er. Math. Astronom. Phys. \textbf{18} (1970), 565--566. \bibitem[J. de Groot et al. \- 1958]{Groot+1958a} J. de Groot and R. J. Wille, \textit{Rigid continua and topological group-pictures}, Arch. Math. (Basel) \textbf{9} (1958), 441--446. \bibitem[Gutek \- 1980]{Gutek1980a} A. Gutek, \textit{Solenoids and homeomorphisms on the Cantor set}, Comment. Math. Prace Mat. \textbf{21} (1980), 299--302. \bibitem[Hagopian \- 1974]{Hagopian1974a} C. L. Hagopian, \textit{$\lambda$-connectivity and mappings onto a chainable indecomposable continuum}, Proc. Amer. Math. Soc. \textbf{45} (1974), 132--136. \bibitem [Hewitt \- 1963]{Hewitt+1963a} E. Hewitt and K. A. Ross, \textit{Abstract harmonic analysis. Vol. I}, Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin-G ttingen-Heidelberg, 1963. \bibitem [Illanes \- 1998]{Illanes1998a} A. Illanes, \textit{A characterization of dendrites with periodic-recurrent property}, Topology Proc. \textbf{23} (Summer 1998), 221--235. \bibitem [Illanes XXXX]{IllanesXXXXa} A. Illanes, \textit{The Buckethandle continuum admits no mean}, preprint. \bibitem [Jel\'\i nek et al. \- 2001]{Jelinek+2001a} P. Jel\'\i nek, P. Podbrdsk\'y and P. Pyrih, \textit{A collection of openly minimal dendrites}, Questions Answers Gen. Topology \textbf{19} (2001), 177--182. \bibitem [Kawamura et al. \- 1996]{Kawamura+1996a} K. Kawamura and E. D. Tymchatyn, \textit{Continua which admit no mean}, Colloq. Math. \textbf{71} (1996), 97--105. \bibitem[Keesling \- 1972]{Keesling1972a} J. Keesling, \textit{The group of homeomorphisms of a solenoid}, Trans. Amer. Math. Soc. \textbf{172} (1972), 119--131. \bibitem[Kennedy Phelps \- 1980]{Kennedy1980a} J. Kennedy Phelps, \textit{Homeomorphisms of products of universal curves}, Houston J. Math. \textbf{6} (1980), 127--134. \bibitem[Krasinkiewicz \- 1969]{Krasinkiewicz1969a} J. Krasinkiewicz, \textit{On homeomorphisms of the Sierpi\'nski curve}, Comment. Math. Prace Mat. \textbf{12} (1969), 255--257. \bibitem[Krasinkiewicz \- 1974a]{Krasinkiewicz1974a} J. Krasinkiewicz, \textit{On a class of indecomposable continua}, Colloq. Math. \textbf{32} (1974), 72--75. \bibitem[Krasinkiewicz \- 1974b]{Krasinkiewicz1974b} J. Krasinkiewicz, \textit{On internal composants of indecomposable plane continua}, Fund. Math. \textbf{84} (1974), 255--263. \bibitem[Krasinkiewicz \- 1976]{Krasinkiewicz1976a} J. Krasinkiewicz, \textit{Mappings onto circle-like continua}, Fund. Math. \textbf{91} (1976), 39--49. \bibitem[Krasinkiewicz \- 1978]{Krasinkiewicz1978a} J. Krasinkiewicz, \textit{Continuous images of continua and 1-movability}, Fund. Math. \textbf{98} (1978), 141--164. \bibitem[Krupski \- 1982]{Krupski1982a} P. Krupski, \textit{A characterization of solenoids-an application of the property of Kelley}, General Topology and its Relations to Modern Analysis and Algebra V, Proc. Fifth Prague Topol. Symp. 1981 (J. Novak, ed.), Heldermann Verlag, Berlin, 1982, 448--449. \bibitem [Krupski \- 1984a]{Krupski1984a} P. Krupski, \textit{Open images of solenoids}, Topology, Proceedings, Leningrad, 1982. Lecture Notes in Mathematics \textbf{1060}, Springer Verlag, 1984, 76--83. \bibitem [Krupski \- 1984b]{Krupski1984b} P. Krupski, \textit{Solenoids and inverse limits of sequences of arcs with open bonding maps}, Fund. Math. \textbf{120} (1984), 41--52. \bibitem[Krupski \- 1984c]{Krupski1984c} P. Krupski, \textit{The property of Kelley, local end-points and homogeneous continua}, Houston J. Math. \textbf{10} (1984), 215-225. \bibitem [Krupski \- 1995]{Krupski1995a} P. Krupski, \textit{The disjoint arcs property for homogeneous curves}, Fund. Math. \textbf{146} (1995), 159--169. \bibitem [Krupski XXXXa]{KrupskiXXXXa} P. Krupski, \textit{Means on solenoids}, Proc. Amer. Math. Soc. (to appear) \bibitem [Krupski XXXXb]{KrupskiXXXXb} P. Krupski, \textit{Hyperspaces of universal curves and $2$-cells are true $F_{\sigma\delta}$-sets}, Colloq. Math., to appear. \bibitem [Krupski XXXXc]{KrupskiXXXXc} P. Krupski, \textit{Families of continua with the property of Kelley, arc continua and curves of pseudo-arcs}, preprint. \bibitem[Krupski et al. XXXXa]{Krupski+XXXXa} P. Krupski and H. Patkowska, \textit{Menger curves in Peano continua}, Colloq. Math. \textbf{70} (1996), 79--86. \bibitem[Krupski et al. XXXXb]{Krupski+XXXXb} P. Krupski and J. T. Rogers, Jr., \textit{Finitely cyclic homogeneous continua}, Proc. Amer. Math. Soc. \textbf{113} (1991), 1173--1177. \bibitem[Kuperberg et al. \- 1995]{Kuperberg+1995a} K. Kuperberg, W. Kuperberg and W. R. R. Transue, \textit{Homology separation and 2-homogeneity}, Continua with the Houston problem book (H. Cook et al., eds.), Dekker, 1995, 287--295. \bibitem [Kuratowski \- 1968]{Kuratowski1968a} K. Kuratowski, \textit{Topology} vol. 2, Academic Press and PWN, New York, London and Warszawa, 1968. \bibitem[Kwapisz \- 2001]{Kwapisz2001a} J. Kwapisz, \textit{Homotopy and dynamics for homeomorphisms of solenoids and Knaster continua}, Fund. Math. \textbf{168} (2001), 251--278. \bibitem[Lelek \- 1961]{Lelek1961a} A. Lelek, \textit{On plane dendroids and their end points in the classical sense}, Fund. Math. \textbf{49} (1961), 301--319. \bibitem[Lewis \- 1987]{Lewis1987a} W. Lewis, \textit{Non-homogeneity of intermediate universal continua}, Topology Proc. \textbf{12} (1987), 193--196. \bibitem[Lum \- 1975]{Lum1975a} L. Lum \citej{A characterization of local connectivity in dendroids}, Studies in topology (Proc. Conf. Univ. North Carolina, Charlotte, NC 1974, Academic Press, New York 1975, pp. 331--338. \bibitem[Ma\'ckowiak \- 1979]{Mackowiak1979a} T. Ma\'ckowiak, \citej{Continuous mappings on continua}, Dissertationes Math. (Rozprawy Mat.)\citen{158} (1979), 1--95. \bibitem[Ma\'ckowiak \- 1984]{Mackowiak1984a} T. Ma\'ckowiak, \textit{Extension theorems for some classes of continua}, Topology Appl. \textbf{17} (1984), 257--263. \bibitem[R. de Man \- 1995]{Man1995a} R. de Man, \textit{On composants of solenoids}, Fund. Math. \textbf{147} (1995), 181--188. \bibitem[Mayer et al. \- 1986]{Mayer+1986a} J. C. Mayer, L. G. Oversteegen and E. D. Tymchatyn, \textit{The Menger curve. Characterization and extension of homeomorphisms of non-locally-separating closed subsets}, Dissertationes Math. (Rozprawy Mat.) \textbf{252} (1986), 1--50. \bibitem[Mazurkiewicz \- 1938]{Mazurkiewicz1938a} S. Mazurkiewicz, \textit{Sur les transformations continues des courbes}, Fund. Math. \textbf{31} (1938), 247--258. \bibitem[Menger \- 1926]{Menger1926a} K. Menger, \textit{Allgemeine R\"aume und Cartesische R\"aume, Zweite Mitteilung: "\"Uber umfassendste $n$-dimensionale Mengen"}, Proc. Akad. Amsterdam \textbf{29} (1926), 1125--1128. \bibitem[Menger \- 1932]{Menger1932a} K. Menger, \textit{Kurventheorie}, Teubner Verlag, Leipzig and Berlin 1932 (reprinted by Chelsea Publ. Co., Bronx, NY 1967). \bibitem[Miller \- 1932]{Miller1932a} E. W. Miller, \textit{Solution of the Zarankiewicz problem}, Bull. Amer. Math. Soc. \textbf{38} (1932), 831--834. \bibitem[Minc \- 1999]{Minc1999a} P. Minc, \textit{Homotopy classes of maps between Knaster continua}, Geometry and topology in dynamics (Winston-Salem, NC, 1998/San Antonio, TX, 1999), Contemp. Math. \textbf{246}, Amer. Math. Soc., Providence, RI, 1999, 189--195. \bibitem[Nadler \- 1977a]{Nadler1977a} S. B. Nadler, Jr., \textit{ Confluent images of the sinusoidal curve}, Houston J. Math. \textbf{3} (1977), 515--519. \bibitem[Nadler \- 1977b]{Nadler1977b} S. B. Nadler, Jr., \textit{ Continua whose cone and hyperspace are homeomorphic}, Trans. Amer. Math. Soc. \textbf{230} (1977), 321--345. \bibitem[Nadler \- 1978]{Nadler1978a} S. B. Nadler, Jr., \citeb{Hyperspaces of sets}, M. Dekker 1978. \bibitem[Nadler \- 1991]{Nadler1991a} S. B. Nadler, Jr., \textit{ Hyperspaces of sets}, M. Dekker, New York and Basel, 1991. \bibitem[Nadler \- 1992]{Nadler1992a} S. B. Nadler, Jr., \textit{ Continuum theory}, M. Dekker, New York, Basel and Hong Kong, 1992. \bibitem[Neumann-Lara et al. \- 1993]{Neumann+1993a} V. Neumann-Lara and I. Puga-Espinosa, \citej{Shore points and dendrites}, Proc. Amer. Math. Soc.\citen{118} (1993), 939--942. \bibitem[Nikiel \- 1983]{Nikiel1983a} J. Nikiel, \textit{A characterizations of dendroids with uncountably many end points in the classical sense}, Houston J. Math. \textbf{9}, (1983), 421--432. \bibitem[Nikiel \- 1985]{Nikiel1985a} J. Nikiel, \textit{On Gehman dendroids}, Glasnik Mat. \textbf{20 (40)} (1985), 203--214. \bibitem[Nikiel \- 1989]{Nikiel1989a} J. Nikiel, \textit{Topologies on pseudo-trees and applications}, Mem. Amer. Math. Soc. \textbf{82} no. 416 (1989), 1--116. \bibitem[Op\v ela et al. \- 2000]{Opela+2000a} D. Op\v ela, P. Pyrih and R. \v S\'amal, \textit{An example of openly minimal dendrite}, Questions Answers Gen. Topology \textbf{18} (2000), 229--236. \bibitem[Oversteegen et al. \- 1994]{Oversteegen+1994a} L. G. Oversteegen and E. D. Tymchatyn, \textit{On the dimension of certain totally disconnected spaces}, Proc. Amer. Math. Soc. \textbf{122} (1994), 885--891. \bibitem[Owens \- 1986]{Owens1986a} M. A. Owens, \citej{Unicoherence at subcontinua}, Topology Appl. \citen{22} (1986), 145--155. \bibitem[Pasynkov \- 1965]{Pasynkov1965a} B A. Pasynkov, \textit{Partial topological products} (Russian), Trudy Moskov. Mat. Ob\v s\v c. \textbf{13} (1965), 136--245. \bibitem[Podbrdsk\'y et al. \- 2001]{Podbrdsky+2001a} P. Podbrdsk\'y and P. Pyrih, \textit{Chain of dendrites without open supremum}, Mathematica Pannonica \textbf{12/2} (2001), 195--200. \bibitem[Prajs \- 1989]{Prajs1989a} J. R. Prajs, \textit{Openly homogeneous continua having only arcs for proper subcontinua}, Topology Appl. \textbf{31} (1989), 133--147. \bibitem[Prajs \- 1998a]{Prajs1998a} J. R. Prajs, \textit{On complete metric spaces containing the Sierpi\'nski curve}, Proc. Amer. Math. Soc. \textbf{126} (1998), 3743--3747. \bibitem[Prajs \- 1998b]{Prajs1998b} J. R. Prajs, \textit{Continuous decompositions of Peano plane continua into pseudo-arcs}, Fund. Math. \textbf{158} (1998) 23--40. \bibitem[Pyrih \- 1999a]{Pyrih1999a} P. Pyrih, \textit{An example of strongly self-homeomorphic dendrite not pointwise self-homeomorphic}, Comment. Math. Univ. Carolin. \textbf{40} (1999), 571--576. \bibitem[Pyrih \- 1999b]{Pyrih1999b} P. Pyrih, \textit{An example related to strongly pointwise self-homeomorphic dendrites}, Arch. Math. (Brno) \textbf{35} (1999), 151--154. \bibitem[Rogers \- 1970]{Rogers1970a} J. W. Rogers, Jr., \textit{On mapping indecomposable continua onto certain chainable indecomposable continua}, Proc. Amer. Math. Soc. \textbf{25} (1970), 449--456. \bibitem[Rogers \- 1971a]{Rogers1971a} J. T. Rogers, Jr., \textit{Embedding the hyperspaces of circle-like plane continua}, Proc. Amer. Math. soc. \textbf{29} (1971), 165--168. \bibitem[Rogers \- 1971b]{Rogers1971b} J. T. Rogers, Jr., \textit{Mapping solenoids onto strongly self-entwined, circle-like continua}, Pacific J. Math. \textbf{37} (1971), 213--216. \bibitem[Rogers \- 1975]{Rogers1975a} J. T. Rogers, Jr., \textit{A cohomological characterization of pre-images of nonplanar, circle-like continua}, Proc. Amer. Math. Soc. \textbf{49} (1975), 232--236. \bibitem[Rogers et al. \- 1971]{Rogers+1971a} J. T. Rogers, Jr. and J. L. Tollefson, \textit{Maps between weak solenoidal spaces}, Colloq. Math. \textbf{23} (1971), 245--249. \bibitem[Sakai \- 1994]{Sakai1994a} K. Sakai, \textit{All autohomeomorphisms of connected Menger manifolds are stable}, Proc. Amer. Math. Soc. \textbf{122} (1994), 1289--1291. \bibitem[Sakai \- 1997]{Sakai1997a} K. Sakai, \textit{Semi-free actions of zero-dimensional compact groups on Menger compacta}, Proc. Amer. Math. Soc. \textbf{125} (1997), 2809--2813. \bibitem[Seaquist \- 1995]{Seaquist1995a} C. R. Seaquist, \textit{A continuous decomposition of the Sierpi\'nski curve}, Continua with the Houston problem book (H. Cook et al., eds.), Dekker, 1995, 315--342. \bibitem[Seaquist \- 1999]{Seaquist1999a} C. R. Seaquist, \textit{Monotone open homogeneity of Sierpi\'nski curve}, Topology Appl. \textbf{93} (1999), 91--112. \bibitem[Sieklucki \- 1959]{Sieklucki1959a} K. Sieklucki, \textit{On a family of power $\mathfrak c$ consisting of $\mathfrak R$-uncomparable dendrites}, Fund. Math. \textbf{46} (1959), 331--335. \bibitem[Sieklucki \- 1961]{Sieklucki1961a} K. Sieklucki, \textit{The family of dendrites $\mathfrak R$-ordered similarly to the segment}, Fund. Math. \textbf{50} (1961), 191--193. \bibitem[Sierpi\'nski \- 1916]{Sierpinski1916a} W. Sierpi\'nski, \textit{Sur une courbe cantorienne qui contient une image biunivoque et continue de toute courbe donne\'e}, C. R. Acad. Sci. Paris \textbf{162} (1916), 629--632. \bibitem[Sierpi\'nski \- 1922]{Sierpinski1922a} W. Sierpi\'nski, \textit{Sur une propri\'et\'e topologique des ensembles fronti\`eres}, Fund. Math. \textbf{3} (1922), 7--13. \bibitem[Smale \- 1965]{Smale1965a} S. Smale, \textit{Diffeomorphisms with many periodic points}, Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), Princeton Univ. Press, Princeton, N.J., 1965, 63--80. \bibitem[\v Stanko \- 1971]{Stanko1971a} M. A. \v Stanko, \textit{Solution of Menger's problem in the class of compacta}, Soviet Math. Dokl. \textbf{12} (1971), 1299--1302. \bibitem[Wa]{Watkins1982a} W. T. Watkins, \textit{Homeomorphic classification of certain inverse limit spaces with open bonding maps}, Pacific J. Math. \textbf{103} (1982), 589--601. \bibitem[Wa\.zewski \- 1923]{Wazewski1923a} T. Wa\.zewski, \textit{Sur les courbes de Jordan ne renfermant aucune courbe simple ferm\'ee de Jordan}, Annales de la Soci\'et\'e Polonaise de Math\'ematique \textbf{2} (1923), 49--170. \bibitem[Whyburn \- 1942]{Whyburn1942a} G. T. Whyburn, \textit{Analytic topology}, Amer. Math. Soc. Colloq. Publ. \textbf{28}, Providence, 1942 (reprinted with corrections 1971). \bibitem[Whyburn \- 1958]{Whyburn1958a} G. T. Whyburn, \textit{Topological chcracterization of the Sierpi\'nski curve}, Fund. Math. \textbf{45} (1958), 320--324. \bibitem[Wilson \- 1972]{Wilson1972a} D. C. Wilson, \textit{Open mappings of the universal curve onto continuous curves}, Trans. Amer. Math. Soc. \textbf{168} (1972), 497--515. \bibitem[Zarankiewicz \- 1925]{Zarankiewicz1925a} K. Zarankiewicz, \textit{Problem 37}, Fund. Math. \textbf{7} (1925), 381. %% %% %%%%%%%%% END REFERENCES %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \end{thebibliography} \end{document}