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\title{Examples in Continuum Theory}
\author{Janusz J. Charatonik, Pawel Krupski and Pavel Pyrih}
\date{ \today}
\maketitle


\tableofcontents


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Here you can find \newtargetlink{source files}{../#1} of this example.
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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.
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Here you can \newtargetlink{ read Notes}{http://www.karlin.mff.cuni.cz/~pyrih/cctbook.html} or
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\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.
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\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}.
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\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
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\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.
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\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
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\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
\e{comb $W_R$}
 and the
\e{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
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\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.
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\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}.
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\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 $D_4$}

\FIGURE[scale=1]{}{../s/c0001/s0020/e0070/}{C}{standard universal dendrite $D_6$}

% Figures 13.3.2, 13.3.3 and 13.3.4 here. ATTENTION: PP is asked to produce
%a new version of Figure 13.3.3 (for $D_4$) by using THIN CONTINUOUS LINES
%as in 13.3.2 and 13.3.4 (NO DOTTED LINES).

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}.
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\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
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\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 $P$}

%PP is able to produce a proper figure and insert it here.
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\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 strongly
pointwise self-homeomorphic dendrite}
\
%PP: please, attach the proper figure here (Fig. 1 on p. 153 of {Pyrih1999b}).

\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}
%PP: attach here, please, your Figure 1 from p. 573 of {Pyrih1999a}.

\rosteree
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\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
$L_0$.
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\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
\e{dendrite $L_0$}
. 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}.
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\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.
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\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
\e{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
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\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}{$Y_3$}

%PP: are you able to produce an illustration of, say, the third
%approximation of $Y$ (i.e., $Y_2$)? It should not be very hard! On maybe
%even $Y_3$??

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}.
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\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
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\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.
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\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
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\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}{$G_4$}


%PP: please, produce a figure of $G_4$. It can be obtained from Fig. 2, p.
%5 of \cite {Arevalo+2001a} by 1) deleting the segment $p^4q$, 2) putting two copies of
%the obtained figure in the same picture (horizontally), and 3) joining the
%two points $p^4$ by a horizontal segment.

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
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\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
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\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}
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\subsection{
{Menger universal continua}
} 
\setcounter{equation}{0}
\label{ex-Menger}

The Menger universal continuum
$M^m_n$, for $1\le n<m$ \cite{Menger1926a},  can
be defined as follows \cite[pp. 121--122]{Engelking1978a}.

 Let $\mathcal F_0=\{I^m\}$.
Inductively, suppose a collection $\mathcal F_k$ of
cubes has been defined for $k\ge 0$. Divide each
cube $D\in\mathcal F_k$  into $3^{m(k+1)}$ congruent cubes
with edges of length $\frac1{3^{m(k+1)}}$. If
$\mathcal F_{k+1}(D)$ is the collection of all these
cubes that intersect $n$-dimensional faces of $D$, then let
$\mathcal F_{k+1}= \bigcup\{\,\mathcal F_{k+1}(D): D\in \mathcal F_k\,\}$.

Define $$M^m_n=\bigcap_{k=0}^\infty \left(\bigcup \mathcal F_k\right).$$

The most popular continua are  $M^2_1$, widely known as the
\gs{Sierpi\'nski universal plane curve}{Sierpinski
universal plane curve} or the \gs{Sierpi\'nski
carpet}{Sierpinski carpet} \cite{Sierpinski1916a}, and
$M^3_1$, called the
\g{Menger universal curve}
 \cite{Menger1926a}. These particular curves
are discussed separately.



 Continuum $M^{2n+1}_n$ can also be obtained as the
 inverse limit $\varprojlim(X_i, p^{i+1}_i)$ in the following way.
 Let  $\{G_1, G_2,\dots\}$ be an open base of $S^n$ consisting of
 open $n$-cells such that $\diam G_i\to 0$. Put $X_1=S^n$ and
 $X_{i+1}=(X_i\times\{0,1\})/\sim$, where relation $\sim$
 identifies points $(x,0),(x,1)$ iff  $p^i_1(x)\notin G_i$.
 The bonding maps $p^{i+1}_i$
are natural retractions \cite{Pasynkov1965a}, \cite[pp. 98--99]{Bestvina1988a}.



\rostere
\item
$M^m_n$ is the \g{universal space} in the
class of all compacta of dimension $\le n$ which embed in
$\mathbb R^m$ \cite{Stanko1971a} (\cite{Menger1926a} for
$M^m_{m-1}$).  $M^m_{m-1}$  is universal in the class of all
$(m-1)$-dimensional subsets of $\mathbb R^m$ (\cite[Problem
1.11.D]{Engelking1978a}), and  $M^{2n+1}_n$ in the class of
all $n$-dimensional separable metric spaces
\cite{Bothe1963a}.
\item\label{ex-Menger-Menger2}
 A continuum $X$ is homeomorphic to $M^{m+1}_m$ if and only
 if $X$ can be embedded in the $(m+1)$-sphere $S^{m+1}$
 in such a way that $S^{m+1}\setminus X$ 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 an
 $m$-cell for each $i$ and $\bigcup_{i=1}^\infty \bd C_i$ is dense in $X$
(see \cite{Cannon1973a}  and \cite[Theorem 6.1.2, p. 74]{Chigogidze+1995a}).
\item\label{ex-Menger-Menger3}
A continuum $X$ is homeomorphic to $M^{2n+1}_n$ if and only
if $X$ is  $n$-dimensional, \gs{$C^{n-1}$}{Ck-space} and
\gs{$LC^{n-1}$-space}{LCk-space} with the
\gs{disjoint $n$-disks property}{disjoint n-disks property}
(\gs{$DD^nP$ property}{DDnP property}) \cite[Corollary 5.2.3,
p.98]{Bestvina1988a}.

Equivalently, $X$ is homeomorphic to $M^{2n+1}_n$ if and
only if $X$ is  $n$-dimensional, $C^{n-1}$ and
$LC^{n-1}$-space such that any continuous map from a compact
space of dimension $\le n$ into   $M^{2n+1}_n$ is a uniform
limit of embeddings (\gs{Z-embeddings}{Z-embedding})
\cite[Theorem 2.3.8, p. 36]{Bestvina1988a}.
\item
\rostere
\item
If $Z\subset M^{2n+1}_n$ is a Z-set\g{Z-set}, then, for
every open neighborhood $U\subset M^{2n+1}_n$ of $Z$ and
every $\epsilon>0$, 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
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\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
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\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}
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%%%%%%%%% START 
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%%%%%%%%% /c0003/s0020/e0050
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%%%%%%%%% 
\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
$$S=\{\,(x,y)\in \mathbb R^2: y=\sin\frac1x, 0<x\le 1\,\}\cup \{\,(0,y):-1\le
y\le 1\,\}.$$
\rostere
\item
$S$ is \g{irreducible} between points $(0,y)$ and
$(1,\sin 1)$, $-1\le y\le 1$. It has exactly three
\gs{end points}{end point} and two arc components.
\item
It is one of the simplest \g{arc-like} continua.
\item
It is a \g{compactification} of a ray
$(0,1]$ with remainder an arc.
\item It has the \g{periodic-recurrent
property} \cite[Corollary 5.10, p. 117]{Charatonik+1997b}.
\item
The only possible \g{confluent} nondegenerate
images of $S$ are an arc and a continuum homeomorphic to $S$
\cite{Nadler1977a}.
\item
The \g{hyperspace} $C(S)$ of all subcontinua of
$S$ is homeomorphic to the \g{cone} over $S$
\cite{Nadler1977b}.
\rosteree

See Figure A.

\FIGURE[scale=1]{}{../s/c0004/s0010/e0010/}{A} {sin curve}

There are many variations of the sin curve. Some of them are pictured below.
See Figure B--C.


\FIGURE[scale=1]{}{../s/c0004/s0010/e0010/}{B}{union of two sin curves}

\FIGURE[scale=1]{}{../s/c0004/s0010/e0010/}{C}{union of two sin curves}
\STORY{../s/c0004/s0010/e0010} 
%%%%%%%%% 
%%%%%%%%% END 
%%%%%%%%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
%%%%%%%%% 
%%%%%%%%% START 
%%%%%%%%% 
%%%%%%%%% /c0004/s0010/e0050
%%%%%%%%% 
%%%%%%%%% 
\subsection{
{Cantor organ and accordion}
} 
\setcounter{equation}{0}

The
\e{Cantor organ}
$X$ is the union of the product $C\times I$ of the Cantor
ternary set $C$ and the unit interval $I$ and  all segments
of the form  $J\times \{0\}$ or $J'\times \{1\}$, where $J$
($J'$, resp.) is the closure of a component of $I\setminus
C$ of length $1/{3^{2k-1}}$ ($1/{3^{2k}}$), $k\in
\mathbb N$ \cite[p. 191]{Kuratowski1968a}. See Figure A.

\FIGURE[scale=1]{}{../s/c0004/s0010/e0050/}{A}{Cantor organ}

\rostere
\item
$X$ is an \g{arc-like} \g{continuum} which is
\g{irreducible} between points $(0,x)$ and $(1,y)$, where
$0\le x,y\le 1$, and  has exactly four
\gs{end points}{end point}.
\item
It has uncountably many arc components.
\rosteree


A variation of the Cantor organ is the
\e{Cantor accordion}
$Y$ which is defined as the \gs{monotone}{monotone map}
image of $X$ under a map that shrinks horizontal bars
$J\times \{0\}$ and $J'\times \{1\}$ to points \cite[p.
191]{Kuratowski1968a}. See Figure B.

\FIGURE[scale=1]{}{../s/c0004/s0010/e0050/}{B}{Cantor accordion}
%Figure as in \cite[p. 191]{Kuratowski1968a} (the right one)

Besides the above properties,
\rosteri
\item
$Y$ has an \g{upper semi-continuous}
\g{monotone decomposition} into arcs with the quotient space
an arc.
\rosterii
\STORY{../s/c0004/s0010/e0050} 
%%%%%%%%% 
%%%%%%%%% END 
%%%%%%%%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\section{
Arc-like (chainable) continua} 
\begin{itemize}
\item [(a)] The simplest arclike continua with $n$ end points
\item [(b)] The pseudo-arc
\end{itemize}
\section{
Irreducible circle-like continua} 
\begin{itemize}
\item [(a)] Solenoids
\item [(b)] The pseudo-circle, pseudo-solenoids
\end{itemize}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
%%%%%%%%% 
%%%%%%%%% START 
%%%%%%%%% 
%%%%%%%%% /c0004/s0030/e0010
%%%%%%%%% 
%%%%%%%%% 
\subsection{
{Solenoids}
} 
\setcounter{equation}{0}

A
\e{solenoid}
is a continuum  homeomorphic to
the inverse limit $\Sigma(\mathbf p)=\varprojlim(S^1,f_n)$
of the inverse sequence of unit circles $S^1$ in the complex
plane with bonding maps $f_n(z)=z^{p_n}$, where $\mathbf
p=(p_1,p_2,\dots)$ is a sequence of prime numbers; it is
called a
\es{$\mathbf p$-adic solenoid}{p-adic solenoid}
.  The solenoid $\Sigma(2,2,\dots)$ is known as a
\e{dyadic solenoid}
.

Geometrically, solenoid  $\Sigma(\mathbf p)$ can be described
as the intersection of a sequence of solid tori $T_1\supset T_2\supset\dots$
such that $T_{n+1}$ wraps $p_n$ times around $T_n$ without
folding and $T_n$ is $\epsilon_n$-thin, for each $n\in \mathbb N$, where
$\lim\epsilon_n=0$.
See Figure A.

\FIGURE[scale=1]{}{../s/c0004/s0030/e0010/}{A} {dyadic selenoid}

\GIF[scale=1]{}{../s/c0004/s0030/e0010/}{AA} {dyadic selenoid - an animation}

%picture of the dyadic solenoid Figure 14.1.3

\rostere
\item
Each solenoid can be constructed (up to a homeomorphism)
as the quotient space of the product $C\times I$ by identifying
each point $(c,1)$ with $(h(c),0)$, where  $h:C\to C$ is
a homeomorphism of the Cantor set $C$ such that
for every $\epsilon>0$ 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 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}
 
\eLIST{}{buckethandle}
\eLIST{}{Cantor accordion}
\eLIST{}{Cantor organ}
\eLIST{}{comb $W$}
\eLIST{}{comb $W_R$}
\esLIST{dendrite $D_m$}{dendrite Dm}
\esLIST{dendrite $D_S$}{dendrite DS}
\eLIST{}{dendrite $L_0$}
\eLIST{}{dendrite of de Groot-Wille type}
\esLIST{dendrite $P$}{dendrite P}
\eLIST{}{double buckethandle}
\eLIST{}{dyadic solenoid}
\eLIST{}{Gehman dendrite}
\eLIST{}{horseshoe}
\eLIST{}{Knaster continuum}
\esLIST{locally connected combs}{locally connected comb}
\eLIST{}{locally connected fan}
\eLIST{}{Menger universal curve}
\eLIST{}{Miller dendrite $S$}
\eLIST{}{modified Miller dendrite}
\esLIST{dendrite $G_{\omega}$}{dendrite Gomega}
\eLIST{}{Omiljanowski dendrite}
\esLIST{$\mathbf p$-adic Knaster continuum}{p-adic Knaster continuum}
\esLIST{$\mathbf p$-adic solenoid}{p-adic solenoid}
\esLIST{Sierpi\'nski carpet}{Sierpinski carpet}
\esLIST{Sierpi\'nski universal plane curve}{Sierpinski universal plane curve}
\eLIST{}{sin curve}
\eLIST{}{solenoid}
\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}
\eSLIST{(topologist's)}{topologist's sin curve}
\esLIST{Wa\.zewski universal dendrite}{Wazewski universal dendrite}
%%%%%%%%% END LIST OF EXAMPLES
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
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%  NEW REFERENCES !!!!!!!!
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% 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.
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