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\subsection{
Sierpi\'nski Universal Plane Curve
}
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The
{{{\it Sierpi\' nski Universal Plane Curve}
\label{Sierpinski Universal Plane Curve}
\index{Sierpinski Universal Plane Curve@Sierpi\' nski Universal Plane Curve}}}
is a well known \g{continuum} which serves as the universal
element in the class of all
\g{one-dimensional} continua in the plane. It is obtained as the
residual set remaining when one begins with a square and
applies the operation of dividing it into nine equal
squares and omitting the interior of the center one, then
repeats this operation on each of the surviving 8 squares,
then repeats again on the surviving 64 squares, and so on
$\cdots$ . See \cite[p.9]{nadler1992}.
The Sierpi\' nski Universal Plane Curve can be characterized
as the only plane \g{locally connected} \g{one-dimensional}
\g{continuum} $S$ such that the boundary of each complementary
domain of $S$ is a \g{simple closed curve} and no two of
these complementary domain boundaries intersect. See
\cite{whyburn1958}.
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\begin{latexonly}
\begin{figure}
\centerline
{{\includegraphics{../s/c0030/s0030/e0020/a.eps}}}
\caption[
(
A
)
Sierpi\'nski Universal Plane Curve
]{
(
A
)
Sierpi\'nski Universal Plane Curve
}
\end{figure}
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\subsection{
Menger's universal curve. Its Anderson's characterization.}
\setcounter{equation}{0}
The \h{Menger Universal Curve} $X$ is obtained using the
\htmlref{Sierpi\' nski Universal Plane Curve} {Sierpinski Universal Plane Curve}
$S$ from the unit cube
$I^3$ by the formula
$$
X = \{(x,y,z) \in I^3 : (x,y) \in S \quad \& \quad(x,z)\in S \quad \& \quad(y,z)\in S\} \quad .
$$
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\addtocounter{figure}{-1}
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\end{rawhtml}
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\begin{latexonly}
\begin{figure}
\centerline
{{\includegraphics{../s/c0030/s0030/e0030/a.eps}}}
\caption[
(
A
)
Menger's universal curve. Its Anderson's characterization.]{
(
A
)
Menger's universal curve. Its Anderson's characterization.}
\end{figure}
\end{latexonly}
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\section{
Universal Menger's compacta. Their Bestvina's characterization. }
The arc-like continua are beautiful
\section{
N\"obeling's examples. (?).}
The arc-like continua are beautiful
\section{
Embeddability of a curve in a surface. Planability. Two Kuratowski's
graphs and two Claytor's curves. Embeddability of a locally connected
continuum in the plane.}
The arc-like continua are beautiful
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%% START of CHAPTER
%%%%%%%%% /c0040
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\chapter{
ORDER OF A POINT}
Last section contains lost continua.
\section{
Menger-Urysohn's order of a point.}
The arc-like continua are beautiful
\section{
Curves of a finite order.}
The arc-like continua are beautiful
\section{
Urysohn's examples: for each positive integer $n \ge 2$ there is a
\cm $X(n)$ consisting solely of points of order $n$ and $2n - 2$.}
The arc-like continua are beautiful
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\subsection{
Sierpi\'nski Tringle
}
\setcounter{equation}{0}
The \hs{Sierpi\'nski Triangle}{Sierpinski Triangle} is
obtained as the residual set remaining when one begins with
a triangle and applies the operation of dividing it into
four equal triangles and omitting the interior of the center
one, then repeats this operation on each of the surviving 3
triangles, then repeats again on the surviving 9 triangles,
and so on $\cdots$ . See \cite[Example 2.7]{cd1994}.
The \gs{Sierpi\'nski Triangle}{Sierpinski Triangle} is
homeomorphic to the unique nonempty compact set $K$ of the
complex plane that satisfies
$$
K= w_1(K) \cup w_2(K) \cup w_3(K) \quad ,
$$
where $w_1$, $w_2$, $w_3$
are maps of the complex plane defined by
$w_1(z)=z/2$, $w_2=(z + 1)/2$ and $w_3=(z+i)/2$.
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\addtocounter{figure}{-1}
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\end{rawhtml}
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\begin{latexonly}
\begin{figure}
\centerline
{{\includegraphics{../s/c0040/s0030/e0010/a.eps}}}
\caption[
(
A
)
Sierpi\'nski Tringle
]{
(
A
)
Sierpi\'nski Tringle
}
\end{figure}
\end{latexonly}
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\section{
Urysohn's examples: for $\mathfrak r \in \{\omega, \aleph_0, \mathfrak c\}$
there is a \cm $X(\mathfrak r)$ consisting solely of points of order $\mathfrak r$}
The arc-like continua are beautiful
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%% /c0050
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\chapter{
DENDRITES}
Last section contains lost continua.
\section{
Dendrites and their basic properties.}
The arc-like continua are beautiful
\section{
Universal dendrites of order $n \ge 3$. Various constructions. }
The arc-like continua are beautiful
\section{
Wa\. zewski's universal dendrite.}
The arc-like continua are beautiful
\section{
Local dendrites.}
The arc-like continua are beautiful
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%
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%%%%%%%%% START of CHAPTER
%%%%%%%%% /c0060
%%%%%%%%%
%%%%%%%%%
\chapter{
INDECOMPOSABILITY}
Last section contains lost continua.
\section{
Early examples.}
The arc-like continua are beautiful
\section{
Common boundary of three or more domains. Examples by R. L. Wilder and M. Luba\'nski. }
The arc-like continua are beautiful
\section{
Indecomposable continua via inverse limits. }
The arc-like continua are beautiful
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%
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%%%%%%%%% START of CHAPTER
%%%%%%%%% /c0070
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\chapter{
IRREDUCIBLE CONTINUA; DECOMPOSITIONS}
Last section contains lost continua.
\section{
???}
The arc-like continua are beautiful
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%% START of CHAPTER
%%%%%%%%% /c0080
%%%%%%%%%
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\chapter{
SPECIAL CONTINUA}
Last section contains lost continua.
\section{
Hereditarily decomposable curves without arcs (Z. Janiszewski).}
The arc-like continua are beautiful
\section{
Whyburn's hereditary cut of the plane. }
The arc-like continua are beautiful
\section{
Hereditarily indecomposable continua. }
The arc-like continua are beautiful
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\subsection{
The pseudoarc}
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Source files:
title.txt .
Here you can
read Notes or
write to Notes.
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\subsection{
The pseudocircle}
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Source files:
title.txt .
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read Notes or
write to Notes.
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\subsection{
$n$-dimensional hereditarily indecomposable continua (R.H.Bing)}
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\section{
The circle of pseudoarcs. }
The arc-like continua are beautiful
\section{
$\Cal P$-like continua. }
The arc-like continua are beautiful
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%
%%%%%%%%% START of CHAPTER
%%%%%%%%% /c0090
%%%%%%%%%
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\chapter{
UNIVERSAL SPACES}
Last section contains lost continua.
\section{
Universal arc-like \cmb (Schori).}
The arc-like continua are beautiful
\section{
Universal circle-like \cmb (Rogers).}
The arc-like continua are beautiful
\section{
Some universal curves (Krasinkiewicz and Minc).}
The arc-like continua are beautiful
\section{
Mohler-Nikiel universal smooth dendroid.}
The arc-like continua are beautiful
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%
%%%%%%%%%
%%%%%%%%% START of CHAPTER
%%%%%%%%% /c0100
%%%%%%%%%
%%%%%%%%%
\chapter{
MAPPINGS}
Last section contains lost continua.
\section{
Mapping properties --- families of continua. }
The arc-like continua are beautiful
\section{
Various kinds of mappings. }
The arc-like continua are beautiful
\section{
Fixed point theory. }
The arc-like continua are beautiful
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%
%%%%%%%%%
%%%%%%%%% START of CHAPTER
%%%%%%%%% /c0110
%%%%%%%%%
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\chapter{
HOMOGENEOUS SPACES}
Last section contains lost continua.
\section{
Homogeneity. }
The arc-like continua are beautiful
\section{
Homogeneous continua. Solenoids. }
The arc-like continua are beautiful
\section{
Homogeneity of Menger compacta (Anderson, W. Lewis).}
The arc-like continua are beautiful
\section{
Variants of homogeneity; Knaster problem; K. Kuperberg's results.}
The arc-like continua are beautiful
\section{
Generalized homogeneity. }
The arc-like continua are beautiful
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%
%%%%%%%%%
%%%%%%%%% START of CHAPTER
%%%%%%%%% /c0120
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\chapter{
HYPERSPACES}
Last section contains lost continua.
\section{
Hyperspaces are terrible ;-)}
The arc-like continua are beautiful
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%%%%%%%%% START of CHAPTER
%%%%%%%%% /c0130
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\chapter{
Lost continua}
This section contains continua from the the first version.
\section{
Arc-like continua }
The arc-like continua are beautiful
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\subsection{
Topologist's Sin Curve}
\setcounter{equation}{0}
The \h {Topologist's Sin Curve} is well known
\g{continuum} which is not \g{arcwise connected}. The continuum is defined
by the union
$$
\{(x,y) \in R^2 : 0 < x \le 1 , y=\sin(1/x)\} \cup
\{(0,y) \in R^2 : -1 \le y \le 1 \} \quad .
$$
It has just two path components. More we can find in
\cite[p.137]{steen1978} .
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\end{rawhtml}
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\begin{latexonly}
\begin{figure}
\centerline
{{\includegraphics{../s/c0130/s0010/e0010/a.eps}}}
\caption[
(
A
)
Topologist's Sin Curve]{
(
A
)
Topologist's Sin Curve}
\end{figure}
\end{latexonly}
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\subsection{
Open Sin}
\setcounter{equation}{0}
The \h{Open Sin} is defined by the Euclidean closure of the
set
$$
\{(x,y) \in R^2 : -1 < x < 1 , y=\sin(1/(x^2-1))\} \quad .
$$
This \g{continuum} has just three path components.
\setcounter{figure}{\value{subsection}}
\addtocounter{figure}{-1}
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\end{rawhtml}
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\begin{latexonly}
\begin{figure}
\centerline
{{\includegraphics{../s/c0130/s0010/e0020/a.eps}}}
\caption[
(
A
)
Open Sin]{
(
A
)
Open Sin}
\end{figure}
\end{latexonly}
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\subsection{
Double Topologists's Sin Curve}
\setcounter{equation}{0}
The \h{Double Topologists's Sin Curve} is defined by the
union
$$
\{(x,y) \in R^2 : 0 < |x| \le 1 , y=\sin(1/x)\} \cup
\{(0,y) \in R^2 : -1 \le y \le 1 \} \quad .
$$
It has just three path components.
\setcounter{figure}{\value{subsection}}
\addtocounter{figure}{-1}
\begin{rawhtml}
\end{rawhtml}
\setcounter{figure}{\value{subsection}}
\addtocounter{figure}{-1}
\begin{latexonly}
\begin{figure}
\centerline
{{\includegraphics{../s/c0130/s0010/e0030/a.eps}}}
\caption[
(
A
)
Double Topologists's Sin Curve]{
(
A
)
Double Topologists's Sin Curve}
\end{figure}
\end{latexonly}
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\subsection{
Snake}
\setcounter{equation}{0}
The \h{Snake} is an \g{arclike} \g{continuum} where each
\g{open} mapping is a homeomorphism.
We can find more in \cite[Theorem ..]{cp1999}.
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\addtocounter{figure}{-1}
\begin{rawhtml}
\end{rawhtml}
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\begin{latexonly}
\begin{figure}
\centerline
{{\includegraphics{../s/c0130/s0010/e0040/a.eps}}}
\caption[
(
A
)
Snake]{
(
A
)
Snake}
\end{figure}
\end{latexonly}
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\subsection{
Cantor Meanders}
\setcounter{equation}{0}
The \h{Cantor Meanders} $X$ is formed by combining the
product $C \times I$ of the \g{Cantor Ternary Set} $C$ and
the unit interval $I$ with the appropriate segments of
$I\times\{0,1\}$. We use the 'zig-zag' system of selecting
the intervals. We start with $[1/3,2/3]\times \{0\}$,
continue with $[1/9,2/9]\times \{1\}$ and $[7/9,8/9]\times
\{1\}$,
$\cdots$ .
The space $X$ has uncountably many \g{arc component}s.
\setcounter{figure}{\value{subsection}}
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\end{rawhtml}
\setcounter{figure}{\value{subsection}}
\addtocounter{figure}{-1}
\begin{latexonly}
\begin{figure}
\centerline
{{\includegraphics{../s/c0130/s0010/e0050/a.eps}}}
\caption[
(
A
)
Cantor Meanders]{
(
A
)
Cantor Meanders}
\end{figure}
\end{latexonly}
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\subsection{
Cantor Snake}
\setcounter{equation}{0}
The \h{Cantor Snake} $X$ is formed by combining the product
$C \times I$ of the \g{Cantor Ternary Set} $C$ and the unit
interval $I$ with countably many \g{Open Sin Curve}s in such
a way that the space $X$ has uncountably many \g{arc components}.
\setcounter{figure}{\value{subsection}}
\addtocounter{figure}{-1}
\begin{rawhtml}
\end{rawhtml}
\setcounter{figure}{\value{subsection}}
\addtocounter{figure}{-1}
\begin{latexonly}
\begin{figure}
\centerline
{{\includegraphics{../s/c0130/s0010/e0060/a.eps}}}
\caption[
(
A
)
Cantor Snake]{
(
A
)
Cantor Snake}
\end{figure}
\end{latexonly}
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\subsection{
Cruller}
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The \h{Cruller} continuum is beautiful.
\setcounter{figure}{\value{subsection}}
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\end{rawhtml}
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\begin{latexonly}
\begin{figure}
\centerline
{{\includegraphics{../s/c0130/s0010/e0080/a.eps}}}
\caption[
(
A
)
Cruller]{
(
A
)
Cruller}
\end{figure}
\end{latexonly}
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\subsection{
Double Buckethandle}
\setcounter{equation}{0}
The \h{Cantor $1/5$-discontinuum} $C_5$ is created using the
procedure of deleting the second and the fourth segments
from five equal segments (instead of the middle third for
the \g{Cantor Ternary Set}). $C_5$ is the set of the reals in
the unit interval which can be represented without $1$-s and
$3$-s in the form
$$
\sum _{n=1} ^\infty \frac{a_n}{5^n} \quad .
$$
The \h{Double Buckethandle} continuum is the union of:
(i) the half circles in the lower plane going
through points of $C_5 \cap \{x : 2/5^{n+1}
\le x \le 1/5^n\}$ with the centre $7/10\cdot 5^n$,
(ii) the half circles in the upper plane going
through points of $C_5 \cap \{x : 2/5^{n+1}
\le 1-x \le 1/5^n\}$ with the centre $1-7/10\cdot
5^n$.
\smallskip
We can find more in \cite[p.205 - p.206]{kuratowski1968}.
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\end{rawhtml}
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\begin{latexonly}
\begin{figure}
\centerline
{{\includegraphics{../s/c0130/s0010/e0090/a.eps}}}
\caption[
(
A
)
Double Buckethandle]{
(
A
)
Double Buckethandle}
\end{figure}
\end{latexonly}
\setcounter{figure}{\value{subsection}}
\addtocounter{figure}{-1}
\begin{latexonly}
\begin{figure}
\centerline
{{\includegraphics{../s/c0130/s0010/e0090/b.eps}}}
\caption[
(
B
)
Double Buckethandle]{
(
B
)
Double Buckethandle}
\end{figure}
\end{latexonly}
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\section{
Self-homeomorphic continua }
Self-homeomorphic continua are beautiful
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\subsection{
Root}
\setcounter{equation}{0}
The \h{Root} is the \g{continuum} obtained attaching
small '$T$' letters to the previous ones as in the figure.
It is a \g{strongly pointwise self-homeomorphic}
\g{dendrite}. It has points of order three on the horizontal
segments and points of order four on the vertical segments.
See \cite[p.232-233]{cd1994}.
\setcounter{figure}{\value{subsection}}
\addtocounter{figure}{-1}
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\end{rawhtml}
\setcounter{figure}{\value{subsection}}
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\begin{latexonly}
\begin{figure}
\centerline
{{\includegraphics{../s/c0130/s0020/e0040/a.eps}}}
\caption[
(
A
)
Root]{
(
A
)
Root}
\end{figure}
\end{latexonly}
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\section{
Dendrites }
Dendrites are beautiful
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\subsection{
Wa\.zewski's Universal Dendrite
}
\setcounter{equation}{0}
The
\hs{Wa\.zewski's Universal Dendrite}{Wazewski's Universal Dendrite}
is well known
\g{continuum} which serves as the universal element in the
class of all \g{dendrite}s. We start with a
\g{Locally Connected Fan} $F_{\omega}$. Denote $D_1=F_{\omega}$. Now we
attach sufficiently small copies of $F_{\omega}$ to all
midpoints of all maximal \g{free arc}s in $D_1$ and obtain a
\g{dendrite} $D_2$. Continuing in this fashion, we obtain the
sequence of \g{dendrite}s $\{D_n\}$ and the
\gs{Wa\.zewski's Universal Dendrite}{Wazewski's Universal Dendrite} $\bf D$ is the closure of their union. For
details see \cite[p.181-185]{nadler1992}.
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\addtocounter{figure}{-1}
\begin{rawhtml}
\end{rawhtml}
\setcounter{figure}{\value{subsection}}
\addtocounter{figure}{-1}
\begin{latexonly}
\begin{figure}
\centerline
{{\includegraphics{../s/c0130/s0030/e0010/a.eps}}}
\caption[
(
A
)
construction of the
Wa\.zewski's Universal Dendrite
]{
(
A
)
construction of the
Wa\.zewski's Universal Dendrite
}
\end{figure}
\end{latexonly}
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\subsection{
Universal Dendrite $D_3$}
\setcounter{equation}{0}
The \h{Universal Dendrite $D_3$}
is obtained by the same procedure as the
\gs{Wa\.zewski's Universal Dendrite}{Wazewski's Universal Dendrite} just instead of locally
connected fans we use simple triods.
We start with a \g{simple triod} $T$. Denote $T_1=T$. Now we
attach sufficiently small copies of $T$ (with one
\g{end point} as the attaching point) to all midpoints of all
maximal \g{free arc}s in $T_1$ and obtain a \g{dendrite}
$T_2$. Continuing in this fashion, we obtain the sequence of
\g{dendrite}s $\{T_n\}$ and the Universal Dendrite $\mathbb D_3$
is the closure of their union.
The Universal Dendrite ${\bf D_3}$ is \g{homogeneous} with
respect to the \g{monotone} mappings. For details see
\cite[p.362]{cc1997}.
\setcounter{figure}{\value{subsection}}
\addtocounter{figure}{-1}
\begin{rawhtml}
\end{rawhtml}
\setcounter{figure}{\value{subsection}}
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\begin{latexonly}
\begin{figure}
\centerline
{{\includegraphics{../s/c0130/s0030/e0020/a.eps}}}
\caption[
(
A
)
Universal Dendrite $D_3$]{
(
A
)
Universal Dendrite $D_3$}
\end{figure}
\end{latexonly}
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\subsection{
Universal Dendrite $D_4$}
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The \h{Universal Dendrite $D_4$} is a
\g{dendtrite} with
\g{ramification point}s of \g{order} 4.
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\begin{figure}
\centerline
{{\includegraphics{../s/c0130/s0030/e0030/a.eps}}}
\caption[
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Universal Dendrite $D_4$]{
(
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Universal Dendrite $D_4$}
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\subsection{
Universal Dendrite $D_6$}
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\h{Universal Dendrite $D_6$} is nice.
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\begin{figure}
\centerline
{{\includegraphics{../s/c0130/s0030/e0040/a.eps}}}
\caption[
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Universal Dendrite $D_6$]{
(
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Universal Dendrite $D_6$}
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\subsection{
Gehman Dendrite}
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The \h{Gehman Dendrite} $G$ is formed from the
\g{Cantor Ternary Set} $C$ in such a way that $C$ is the set of
\g{end point}s of $G$ and all \g{ramification point}s of $G$ are of
\g{order} three.
See \cite[pp. 422-423]{nikiel1983} and \cite[p. 203]{nikiel1985}.
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\begin{figure}
\centerline
{{\includegraphics{../s/c0130/s0030/e0050/a.eps}}}
\caption[
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Gehman Dendrite]{
(
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Gehman Dendrite}
\end{figure}
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\section{
Dendroids }
Dendroids are beautiful
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\subsection{
Cantor Comb}
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The \h{Cantor Comb} $X$ is formed from the \g{Cantor Ternary Set}
$C$ and the unit interval $I$ using the formula
$$
X= C \times I \quad \cup \quad I \times\{0\} \quad .
$$
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\begin{figure}
\centerline
{{\includegraphics{../s/c0130/s0040/e0010/a.eps}}}
\caption[
(
A
)
Cantor Comb]{
(
A
)
Cantor Comb}
\end{figure}
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\subsection{
Gehman Dendrite with Ski}
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The \h{Gehman Dendrite with Ski} $G_s$ is a \g{dendroid}
formed from the \g{Gehman Dendrite} $G$ and the
\g{Cantor Ternary Set} $C$ with a formula
$$
G_s = \{(x,y,z)\in R^3 : (x,y)\in G \} \cup
\{(x,y,z)\in R^3 : x\in C, y=0, |z|\le 1 \} \quad .
$$
In fact we attach uncountably many segments of length 2 with
their midpoints to the points of \g{Cantor Ternary Set} (the
\g{end point}s of \g{Gehman Dendrite}). See \cite[p. 16]{jjc1995}.
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\begin{figure}
\centerline
{{\includegraphics{../s/c0130/s0040/e0020/a.eps}}}
\caption[
(
A
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Gehman Dendrite with Ski]{
(
A
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Gehman Dendrite with Ski}
\end{figure}
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\subsection{
Makuchowski Twin}
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We obtain the \h{Makuchowski Twin} as a subset of the
Euclidean plane with the induced topology by attaching the
\g{Locally Connected Fan}
to the \g{end point} of the prolonged segment of the
\g{Prolonged Harmonic Fan}.
Makuchowski twin is a \g{dendroid} where each nonconstant
\g{open} mapping is \g{light}. See \cite[Example 2.1]{makuchowski1994}.
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\begin{figure}
\centerline
{{\includegraphics{../s/c0130/s0040/e0030/a.eps}}}
\caption[
(
A
)
Makuchowski Twin]{
(
A
)
Makuchowski Twin}
\end{figure}
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\section{
Fans }
Fans are beautiful
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\subsection{
Harmonic Fan}
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We define the \h{Harmonic Fan}
$F_{H}=\cup \{ve : e \in H\}$, where $ve$ is the
segment joining $v=(0,1)$ with
$e\in H$, $H=\{0\}\cup\{1/n:n\in N\}$.
We consider $F_{H}$ as a subset of the
Euclidean plane with the induced topology.
See \cite[p.9]{ccm1990}.
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\begin{figure}
\centerline
{{\includegraphics{../s/c0130/s0050/e0010/a.eps}}}
\caption[
(
A
)
Harmonic Fan]{
(
A
)
Harmonic Fan}
\end{figure}
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\subsection{
Harmonic Shredded Fan}
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To define the \g{Harmonic Shredded Fan} $F_{HS}$ we cut the
'arms' $ve$ in the \g{Harmonic Fan} $F_{H}$ in such a way,
that their lengths run through all rationals in the unit
interval (we do not cut the vertical segment through the
origin). We notice that all Harmonic shredded fans are
homeomorphic.
See \cite[p.31]{ccm1990}.
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\begin{figure}
\centerline
{{\includegraphics{../s/c0130/s0050/e0020/a.eps}}}
\caption[
(
A
)
Harmonic Shredded Fan]{
(
A
)
Harmonic Shredded Fan}
\end{figure}
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\subsection{
Cantor Fan}
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We define the \h{Cantor Fan}
$F_{C}=\cup \{ve : e \in C\}$, where $ve$ is the
segment joining $v=(1/2,1)$ with all points
$e$ in the \g{Cantor Ternary Set} $C$.
We consider $F_{C}$ as a subset of the
Euclidean plane with the induced topology.
See \cite[p.9]{ccm1990}.
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\begin{figure}
\centerline
{{\includegraphics{../s/c0130/s0050/e0030/a.eps}}}
\caption[
(
A
)
Cantor Fan]{
(
A
)
Cantor Fan}
\end{figure}
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\subsection{
Lelek Fan}
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We define the \h{Lelek Fan}
$F_{L}$ a subfan of the \g{Cantor Fan} $F_C$ such that
the set $E(F_L)$ of its \g{end point}s is dense in $F_L$,
and $S(F_L)=\{v\} \cup E(F_L)$ is a connected set (where $v$
denotes the \g{top} of the two \g{fan}s $F_C$ and $F_L
\subset F_C$).
It is known, that any two \g{Lelek Fan}s are homeomorphic.
See \cite[p.9]{ccm1990} . Moreover the \g{Lelek Fan} is
homeomorphic to all its \g{confluent} images (see
\cite[]{wjc1989}).
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\end{rawhtml}
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\begin{figure}
\centerline
{{\includegraphics{../s/c0130/s0050/e0040/a.eps}}}
\caption[
(
A
)
Lelek Fan]{
(
A
)
Lelek Fan}
\end{figure}
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\section{
Planable continua }
Planable continua are beautiful
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\subsection{
Makuchowski Umbrella}
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\h{Makuchowski Umbrella} is a \g{hereditarily} locally
connected \g{continuum} where each nonconstant \g{open}
mapping is \g{light}. This property disappears when we cut
out those vertical segments due to the horizontal projection
on appropriate vertical segment.
See \cite[Example 2.5]{makuchowski1994}.
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\begin{figure}
\centerline
{{\includegraphics{../s/c0130/s0060/e0010/a.eps}}}
\caption[
(
A
)
Makuchowski Umbrella]{
(
A
)
Makuchowski Umbrella}
\end{figure}
\end{latexonly}
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\chapter{
New examples}
New section contains new examples.
\section{
New examples }
New examples are nice.
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\subsection{
Whyburn's Curve}
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The well-known Jordan Curve Theorem says that a simple closed curve in the
plane cuts the plane into two regions and is their common boundary. Thus, it
is a cutting of the plane. The so called Warsaw Circle, obtained from the
$\sin (1/x)$-curve
$$S = \{(0,y) \in \Bbb R^2: y \in [-1,1]\} \cup \{(x,\sin(1/x))
\in \Bbb R^2: x \in (0,1] \}$$
by identifying the points $(0,-1)$ and $(1, \sin 1)$ also is
a cutting. But both these continua contain arcs, which
certainly do not cut the plane. So, one can ask a question
if there exists such a cutting $X \subset \Bbb R^2$ that
each nondegenerate subcontinuum of $X$ also cuts $\Bbb R^2$.
The question has been answered in the affirmative in 1930 by
G. T. Whyburn who constructed in [3] a {\it hereditary
cutting of the plane}, which is now called the \h{Whyburn's
Curve}. The present description of this peculiar continuum
is a modification of the original one due to A. Lelek [1],
to obtain some extra properties of the curve, and it follows
Lelek's sketch of the construction in [3, p. 117-119]. We
say that a continuum $X$ lying in the plane $\Bbb R^2$ cuts
the plane between points $a$ and $b$ if the points belong to
different components of $\Bbb R^2 \setminus X$. A continuum
$X$ is a cutting of $\Bbb R^2$ if there are some two points
such that $X$ cuts $\Bbb R^2$ between them. Take the
$\sin(1/x)$curve $S$ and identify points $(0,1)$ and
$(0,-1)$ into one point $r$. The obtained curve (still
considered as a subset of the plane) consists of a circle
$C$ and a sinusoidal curve approximating the circle, so it
is a cutting of the plane. If we add to the curve an arc
starting at $r$ and having only this point in common with
the curve, we get a continuum $Y$ being also a cutting of
the plane (namely it cuts the plane between a point $a$
inside the circle $C$ and a point $b$ lying in $\Bbb R^2
\setminus Y$ outside $C$) and having the property that every
subcontinuum of $Y$ containing a point $p$ of the sinusoidal
part of $Y$ and a point $q$ lying on the added arc, also
cuts the plane between $a$ and $b$ (see Figure A (I)-(II).).
\par The same curve $Y$ can also be obtained as the
intersection of a nested sequence of plane continua. Each
continuum of the sequence has the form of a long strip with
one loop and finitely many zig-zags. Consecutive strips are
thinner and thinner, more and more narrow, each next being
inscribed in the previous one in such a way that the next
loop is contained in the previous loop which also contains
one more zig-zag of the next strip (see Figure A
(III)-(IV)). Consequently, the number of zig-zags in the
next approximation is greater than that in the previous one,
so these numbers tend to infinity. The common part of the
all strips is just the curve $Y$, while the common part of
the loops is the circle $C$ contained in $Y$. \par
The Whyburn's curve $W$ can be obtained in a similar way, as
the intersection of a nested sequence of strips in the
plane, using the method called "condensation of
singularities". Namely the singularity appearing in each
neighborhood of $C$ in $Y$ is copied more and more times
both in the loops as well as in the zig-zag parts of the
consecutive strips. A little bit more precisely speaking,
the strips are modified so that each next strip, magnifying
the number of zig-zags in the loops and preserving previous
loops, contains also new, smaller loops, distributed in a
more and more dense manner inside the previous strip (see
Figure A (V)) so that the diameters of the loops tend to
zero as the number of steps of the construction tends to
infinity. As a consequence of this property, for every two
points $p$ and $q$ of the curve $W$ one can find, in a
sufficiently far approximating strip, a loop lying between
these points. Roughly speaking, behavior of the curve $W$
between $p$ and $q$ is such as that of the curve $Y$, i.e.,
each subcontinuum of $W$ containing $p$ and $q$ cuts the
plane between some points $a$ and $b$ determined by the
mentioned loop. \par The continuum $W$ constructed is this
way has, among others, the following properties (see [1]).
1) each subcontinuum of $W$ is a cutting of the plane, and
therefore $W$ contains no arc; 2) each subcontinuum of $W$
contains a homeomorphic copy of $W$; 3) $W$ is hereditarily
decomposable; and 4) $W$ is a continuous image of the
pseudo-arc $\Bbb P$. Since neither the pseudo-arc $\Bbb P$
nor any subcontinuum of $\Bbb P$ cuts the plane, this last
property shows that a hereditary anti-cutting of $\Bbb R^2$
can be continuously mapped onto a hereditary cutting.
\medskip
[1]
A. Lelek : On weakly chainable continua, Fund. Math. 51
(1962), 271--282.
[2] A. Lelek : Zbiory, PZWS, Warszawa 1966.
[3] G. T. Whyburn : A continuum every subcontinuum of which
separates the plane, Amer. J. Math. 52 (1930), 319--330.
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\begin{figure}
\centerline
{{\includegraphics{../s/c0140/s0010/e0010/a.eps}}}
\caption[
(
A
)
construction of the
Whyburn's Curve]{
(
A
)
construction of the
Whyburn's Curve}
\end{figure}
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\subsection{
Pseudo-arc}
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A. Lelek, ZBIORY (Sets), PZWS Warszawa 1966, pages
90-95.\bigskip The first hereditarily indecomposable
continuum has been constructed by B. Knaster in 1922. This
continuum $K$, discovered by Knaster, is just named the {\it
pseudo-arc} or {\it Knaster's continuum}. The pseudo-arc $K$
is obtained as the common part of a decreasing sequence of
continua in the plane, every of which is the union of
finitely many discs, forming a chain of sets. \medskip We
will consider chains
$$\Cal D = (D_1, D_2, \dots, D_k)$$
of discs $D_i$ in the plane, having the property that any
two consecutive disc $D_i$ and $D_{i+1}$ have interior
points in common, i.e., that their intersection --- nonempty
by assumption --- is not contained in the boundary of any of
them (Figure A). Discs $D_i$ are called {\it links} of the
chain $\Cal D$. The union of all links of the chain $\Cal D$
is a continuum. Observe that if we delete from the chain
$\Cal D$ a link which is not an end link, i.e., a link $D_i$
such that $1 < i < k$, then the union of the remaining links
is a not connected set, having two components. \par Let two
chains of discs $\Cal D$ and $\Cal D'$ be given. We say that
the chain $\Cal D'$ {\it refines} the chain $\Cal D$ if each
link $D'_s$ of the chain $\Cal D'$ is contained in the
interior of at least one link $D_i$ of the chain $\Cal D$,
i.e., if for every $D'_s \in \Cal D'$ there exists a link
$D_i \in \Cal D$ such that $D'_s \subset D_i$ and that the
disc $D'_s$ has no common points with the boundary of the
disc $D_i$ (Figure B). \par We say that the chain $\Cal D'$
is {\it crooked} in the chain $\Cal D$ if it refines $\Cal
D$ and if, for every pair of links $D_i$ and $D_j$ of the
chain $\Cal D$ such that
\begin{equation}
i + 2 < j,
\end{equation}
and for every pair of links $D'_s$ and $D'_v$ of the chain
$\Cal D'$, intersecting the links $D_i$ and $D_j$
respectively, i.e.,
$$D_i \cap D'_s \ne \emptyset \ne D_j \cap D'_v,$$
there are, in the chain $\Cal D'$, links $D'_t$ and $D'_u$, lying between
the link $D'_s$ and $D'_v$ in the same order, i.e.,
$$s < t < u < v \quad \mbox { or } \quad s > t > u > v,$$
such that
\begin{equation}
D'_t \subset D_{j-1} \quad \mbox { and } \quad D'_u
\subset D_{i+1}.
\end{equation}
In other words, a part of the chain $\Cal D'$ which is
contained between the links $D'_s$ and $D'_v$ has to form a
fold in the chain $\Cal D$ between the links $D_i$ and $D_j$
(Figure C). The chain $\Cal D'$, to go from the link $D_i$
to the link $D_j$ has first to come to the link $D_{j-1}$,
next go back to the link $D_{i+1}$, and only after this it
can reach the link $D_j$. Such a fold must exist for every
two links $D_i$ and $D_j$ having no adjoining (i.e.
neighbor) link in common. \medskip Now, let there be given
in the plane two points $a$ and $b$, and an infinite
sequence
$$\Cal D_1, \Cal D_2, \dots , \Cal D_n, \dots $$
of chains of discs, satisfying, for every $n \in \{1, 2, \dots \}$, the
following conditions:
\begin{description}
\item [$~~~1^o$] the point $a$ belongs to the first, and the point $b$ to the
last link of the chain $\Cal D_n$;
\item [$~~~2^o$] the diameter of every disc in the chain $\Cal D_n$ is less
than $\frac 1n$;
\item [$~~~3^o$] the chain $\Cal D_{n+1}$ is crooked in the chain $\Cal D_n$.
\end{description}
\par Thus, for example, the chain $\Cal D_2$ is crooked in the chain
$\Cal D_1$, and the chain $\Cal D_3$ is crooked in the chain $\Cal D_2$ (on
Figure D only a part of the chain $\Cal D_3$ is presented). The foldings are
more and more condensed and they overlapped themselves. \medskip
Denote by $K_n$ the union of all links of the chain $D_n$. From condition
$3^o$ it follows in particular that the each next chain refines the previous
one, and therefore the continua $K_n$ form a decreasing sequence
$$K_1 \supset K_2 \supset \dots \supset K_n \supset \dots $$
\par The \h{Pseudo-arc} $K$ is defined as the common part of all continua
$K_n$, i.e.,
$$K = \bigcap \{K_n: n \in \Bbb N\}.$$
By virtue of condition $1^o$ points $a$ and $b$ belong to the continuum $K$,
thus it is nondegenerate. We shall prove that the continuum $K$ is
hereditarily indecomposable.\medskip
Let $K' \subset K$ be an arbitrary continuum. To prove that the continuum
$K$ is indecomposable it is enough to show that every continuum $Y
\subset K'$ distinct from $K$' is nowhere dense in $K'$. Consider an
arbitrary point $p \in Y$ and number $\eps > 0$. \par
Since $K' \ne Y$, hence there is a point $q \in K' \setminus Y$ which
therefore is at a distance at least $\eta > 0$ from each point $y \in Y$,
where $\eta$ is a fixed number, independent from $y$. In fact, if not, then
some points of the set would be arbitrarily close to the point $q$, and so
the point $q$ would belong to the closure of the set $Y$, which is
impossible, because the set $Y$ is closed, and it does not contain the point
$q$. Let us take a natural number $n$ so large that the inequalities are
satisfied
$$ \frac 2n < \eps \quad \mbox { and } \quad \frac 3n < \eta.$$ \par
Denote by $D_i$ and $D_j$ (with $i \le j$) the links of the chain $\Cal D_n$
whose union contains the points $p$ and $q$. Since $p \in Y$, hence
$\rho (p,q) \ge \eta$, (here $\rho$ means the metric in the plane) and thus
it follows that between the links $D_i$ and $D_j$ there are at least two
links of the chain $\Cal D_n$. Indeed, in the opposite case the links $D_i$
and $D_j$ would have a neighbor common link, and choosing suitable points
$p'$ and $q'$ in the intersections of this link with the links $D_i$ and
$D_j$ we would have
$$
\rho (p,q) \le \rho (p,p') + \rho (p',q') + \rho (q',q) < \frac 1n +
\frac 1n + \frac 1n = \frac 3n < \eta
$$
by condition $2^o$. Thus we can assume that the indices $i$ and $j$ satisfy
the inequality (1). Further, we can assume also that the point $p$ belongs
to one, and the point $q$ to the other of the links $D_i$ and $D_j$. \par
Denote by $D'_s$ and $D'_v$ the links of the chain $\Cal D_{n+1}$ which
contain points $p$ and $q$, respectively. Additionally we assume that
$p \in D_i$. In the opposite case, i.e. if $p \in D_j$, the further part of
the proof runs in the same way (with changing of the roles of the links
$D_i$ and $D_j$ as well as $D_{i+1}$ and $D_{j-1}$). \par
So,
$$p \in D_i \cap D'_s \quad \mbox { and } \quad q \in Dj \cap D'_v,$$
whence we infer, by virtue of condition $3^o$ and the definition of folding
of chains, that there are two links $D'_t$ and $D'_u$ in the chain $\Cal
D_{n+1}$ which lie between the links $D'_s$ and $D'_v$ in the same order and
which satisfy the inclusions (2). The union of the links of the chain
$\Cal D_{n+1}$ distinct from the link $D'_u$ is a not connected set (see
above) containing the point $p$ in one component, and the point $q$ in the
other. Since the continuum $K'$ joins the points $p$ and $q$ and is
contained in the union $K_{n+1}$ of all links of the chain $\Cal D_{n+1}$,
hence $K'$ must pass thru the link $D'_u$. Thus there is a point
$$x \in K' \cap D'_u$$
(Figure E). We shall prove that the point $x$ does not belong to the
continuum $Y$. Indeed, if it would be $x \in Y$, then by the same reason as
previously for the continuum $K'$ the continuum $Y$ would have to pass thru
the link $D'_t$. Then there would exist a point
$$y \in Y \cap D'_t,$$
which would be in the link $D_{j-1}$ according to (2). But
the neighbor links $D_j$ and $D_{j-1}$ have their diameters
less than $\frac 1n$ by condition $2^o$, and therefore we
would have
$$\rho (y,q) < \frac 2n < \frac 3n < \eta,$$
which is impossible because $y \in Y$. \par So, the point
$x$ belongs to the set $K' \setminus Y$ and to the link
$D_{i+1}$ simultaneously. The neighbor links $D_i$ and
$D_{i+1}$ have diameters less than $\frac 1n$, whence it
follows that
$$\rho (p,x) < \frac 2n < \eps.$$ \par
Since $\eps$ was an arbitrary positive number, hence, in
this way we have proved in that there are points of the set
$K' \setminus Y$ which lie arbitrarily closely to the point
$p$. The point $p$ was an arbitrary point of the continuum
$Y$. Thus we have proved that the continuum $Y$ is nowhere
dense in $K'$, and the proof of hereditary indecomposability
of the continuum $K$ is finished.
An example of a chain crooked in a chain with 7 links is on
Figure F.
\setcounter{figure}{\value{subsection}}
\addtocounter{figure}{-1}
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\end{rawhtml}
\setcounter{figure}{\value{subsection}}
\addtocounter{figure}{-1}
\begin{latexonly}
\begin{figure}
\centerline
{{\includegraphics{../s/c0140/s0010/e0020/a.eps}}}
\caption[
(
A
)
a chain with links, construction of the
Pseudo-arc]{
(
A
)
a chain with links, construction of the
Pseudo-arc}
\end{figure}
\end{latexonly}
\setcounter{figure}{\value{subsection}}
\addtocounter{figure}{-1}
\begin{latexonly}
\begin{figure}
\centerline
{{\includegraphics{../s/c0140/s0010/e0020/b.eps}}}
\caption[
(
B
)
refining a chain, construction of the
Pseudo-arc]{
(
B
)
refining a chain, construction of the
Pseudo-arc}
\end{figure}
\end{latexonly}
\setcounter{figure}{\value{subsection}}
\addtocounter{figure}{-1}
\begin{latexonly}
\begin{figure}
\centerline
{{\includegraphics{../s/c0140/s0010/e0020/c.eps}}}
\caption[
(
C
)
crookedness used in the construction of the
Pseudo-arc]{
(
C
)
crookedness used in the construction of the
Pseudo-arc}
\end{figure}
\end{latexonly}
\setcounter{figure}{\value{subsection}}
\addtocounter{figure}{-1}
\begin{latexonly}
\begin{figure}
\centerline
{{\includegraphics{../s/c0140/s0010/e0020/d.eps}}}
\caption[
(
D
)
crookedness used in the construction of the
Pseudo-arc]{
(
D
)
crookedness used in the construction of the
Pseudo-arc}
\end{figure}
\end{latexonly}
\setcounter{figure}{\value{subsection}}
\addtocounter{figure}{-1}
\begin{latexonly}
\begin{figure}
\centerline
{{\includegraphics{../s/c0140/s0010/e0020/e.eps}}}
\caption[
(
E
)
crookedness used in the construction of the
Pseudo-arc]{
(
E
)
crookedness used in the construction of the
Pseudo-arc}
\end{figure}
\end{latexonly}
\setcounter{figure}{\value{subsection}}
\addtocounter{figure}{-1}
\begin{latexonly}
\begin{figure}
\centerline
{{\includegraphics{../s/c0140/s0010/e0020/f.eps}}}
\caption[
(
F
)
an example of a chain crooked in a chain with 7 liks, construction of the
Pseudo-arc]{
(
F
)
an example of a chain crooked in a chain with 7 liks, construction of the
Pseudo-arc}
\end{figure}
\end{latexonly}
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\subsection{
Dyadic Selenoid}
\setcounter{equation}{0}
The \h{Dyadic Selenoid} is obtained easily :-)
\setcounter{figure}{\value{subsection}}
\addtocounter{figure}{-1}
\begin{rawhtml}
\end{rawhtml}
\setcounter{figure}{\value{subsection}}
\addtocounter{figure}{-1}
\begin{latexonly}
\begin{figure}
\centerline
{{\includegraphics{../s/c0140/s0010/e0030/a.eps}}}
\caption[
(
A
)
first step in the construction of the
Dyadic Selenoid]{
(
A
)
first step in the construction of the
Dyadic Selenoid}
\end{figure}
\end{latexonly}
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\subsection{
Zig-zag Circle}
\setcounter{equation}{0}
The \h{Zig-zag Circle} is obtained easily :-)
\setcounter{figure}{\value{subsection}}
\addtocounter{figure}{-1}
\begin{rawhtml}
\end{rawhtml}
\setcounter{figure}{\value{subsection}}
\addtocounter{figure}{-1}
\begin{latexonly}
\begin{figure}
\centerline
{{\includegraphics{../s/c0140/s0010/e0040/a.eps}}}
\caption[
(
A
)
Zig-zag Circle]{
(
A
)
Zig-zag Circle}
\end{figure}
\end{latexonly}
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\subsection{
M-Continuum}
\setcounter{equation}{0}
The \h{M-Continuum} is obtained easily :-)
\setcounter{figure}{\value{subsection}}
\addtocounter{figure}{-1}
\begin{rawhtml}
\end{rawhtml}
\setcounter{figure}{\value{subsection}}
\addtocounter{figure}{-1}
\begin{latexonly}
\begin{figure}
\centerline
{{\includegraphics{../s/c0140/s0010/e0050/a.eps}}}
\caption[
(
A
)
M-Continuum]{
(
A
)
M-Continuum}
\end{figure}
\end{latexonly}
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\subsection{
Buckethandle}
\setcounter{equation}{0}
The \h{Buckethandle} is created from the
\g{Cantor Ternary Set} $C$ with this procedure:
(i) we join any two points $a$ and $b$ in $C$
symmetric with respect to $1/2$ with a semicircle in
the upper half plane with the centre in $(1/2,0)$,
(ii) we join any two points of $C$ in the interval
$2/3^n \le x \le 3/3^n$, $n\ge 1$, with a semicircle
with the centre in $(5/(2\cdot 3^n),0)$ in the lower
half plane.
This continuum is often called the
\h{Knaster's Buckethandle} continuum
(see Figure (A)).
\medskip
This kind of curiosities in continuum theory is connected
with common boundary of several plane regions. In 1904 A.
Sch\"onflies started to publish a sequence of papers
[6] which became an important step in development of
continuum theory. Relying heavily on intuition, he claimed
that there do not exist three regions in the plane with
common boundary. The claim was refused by L. E. J. Brouwer
in 1910 [1] who constructed continua which are common
boundary of three regions and showed that they are {\it
indecomposable}, i.e., such continua $X$ that there are no
nonempty proper subcontinua $A$ and $B$ of $X$ with $X = A
\cup B$. The first example of such a continuum was given by
Brouwer in 1910 (see Figure B, where the first several steps
of his construction, simplified by Z. Janiszewski
(1888--1920) [2, p. 114] are presented). Finally in
1922 B. Knaster (1893--1980) gave a nice description of this
continuum (with the first full proof of its
indecomposability) in [3, p. 209-210]. Now the
continuum is referred to as the simplest indecomposable
continuum, or the horse-shoe continuum, or the B-J-K
continuum (for Brouwer, Janiszewski and Knaster). \par
Finally the common boundary problem of plane domains has
been solved in 1928 by K. Kuratowski (1896--1980) who proved
in [4] and [5] that every plane continuum
which is the common boundary of $n$ open domains either is
indecomposable or is the union of two indecomposable
continua whenever $n
\ge 3$, and when $n = 2$ it either is "monostratic" or has a natural "cyclic
structure" in the sense that it is built up from layers naturally ordered in
the same way as the individual points of the circle. \par
\smallskip
We can obtain a homeomorphic copy of the buckethandle
continuum using the \g{inverse limit}
$$
\lim \limits _{\leftarrow} \{X_i,f_i\}_{i=1}^\infty
$$
where for each $i=1,2,\cdots $ let $X_i=[0,1]$ and
$f_i(t)=2t$ for $0\le t \le 1/2$ and $f_i(t)=-2t+2$
for $1/2\le t \le 1$.
\smallskip
We can find more in \cite[p.22]{nadler1992} and
\cite[p.205]{kuratowski1968}.
[1] L. E. J. Brouwer : Zur Analysis Situs, Math. Ann.
68(1910), 422--434.
[2] Z. Janiszewski: Sur les continus irr\'eductibles entre deux points,
Journal de l'Ecole Polytechnique (2) 16(1912), 79--170.
[3] K. Kuratowski: Th\'eorie des continus irr\'eductibles entre deux points
I, Fund. Math. 3(1922), 200--231.
[4] K. Kuratowski: Sur les coupures irr\'eductibles du
plan, Fund. Math. 6(1924), 130--145.
[5] K. Kuratowski: Sur la structure des fronti\`eres communes \`a deux
regions, Fund. Math. 12(1928), 20--42.
[6] A. Sch\"onflies: Beirtr\"age zur Theorie der Punktmengen
{I}, Math. Ann. 58(1904), 195--244;
{II} 59(1904), 129--160;
{III} 62(1906), 286--236.
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\addtocounter{figure}{-1}
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\end{rawhtml}
\setcounter{figure}{\value{subsection}}
\addtocounter{figure}{-1}
\begin{latexonly}
\begin{figure}
\centerline
{{\includegraphics{../s/c0140/s0010/e0060/a.eps}}}
\caption[
(
A
)
Buckethandle]{
(
A
)
Buckethandle}
\end{figure}
\end{latexonly}
\setcounter{figure}{\value{subsection}}
\addtocounter{figure}{-1}
\begin{latexonly}
\begin{figure}
\centerline
{{\includegraphics{../s/c0140/s0010/e0060/b.eps}}}
\caption[
(
B
)
several steps in the construction of the
Buckethandle]{
(
B
)
several steps in the construction of the
Buckethandle}
\end{figure}
\end{latexonly}
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%%%%%%%%% /c0140/s0010/e0070
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\subsection{
Cantor Interaction}
\setcounter{equation}{0}
\h{Cantor Interaction} is a Charatonic's arcwise connected world ...
\setcounter{figure}{\value{subsection}}
\addtocounter{figure}{-1}
\begin{rawhtml}
\end{rawhtml}
\setcounter{figure}{\value{subsection}}
\addtocounter{figure}{-1}
\begin{latexonly}
\begin{figure}
\centerline
{{\includegraphics{../s/c0140/s0010/e0070/a.eps}}}
\caption[
(
A
)
Cantor Interaction]{
(
A
)
Cantor Interaction}
\end{figure}
\end{latexonly}
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%%%%%%%%% /c0140/s0010/e0080
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\subsection{
Pseudo-circle}
\setcounter{equation}{0}
\h{Pseudo-circle} is a strange \g{continuum}.
\setcounter{figure}{\value{subsection}}
\addtocounter{figure}{-1}
\begin{rawhtml}
\end{rawhtml}
\setcounter{figure}{\value{subsection}}
\addtocounter{figure}{-1}
\begin{latexonly}
\begin{figure}
\centerline
{{\includegraphics{../s/c0140/s0010/e0080/a.eps}}}
\caption[
(
A
)
A circular chain crooked in a circular chain with 6 links, construction of the
Pseudo-circle]{
(
A
)
A circular chain crooked in a circular chain with 6 links, construction of the
Pseudo-circle}
\end{figure}
\end{latexonly}
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%%%%%%%%% /c0140/s0010/e0090
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\subsection{
Semicircle Continuum}
\setcounter{equation}{0}
\h{Semicircle Continuum} is a nice \g{continuum}.
Let $A = \{(x,0)\in \Bbb R^2: 0 \le x \le 1\}$;
for each $n=1,2,\cdots$ and each $k=1,\cdots, 2^{n-1}$, let
$$
B_{n,k} = \left \{(x,y)\in \Bbb R^2:
\left (x-\frac{2k-1}{2^n}\right )^2 + y^2
=4^{-n} \mbox{and} y\ge 0\right \};
$$
for each $n=0,1,\cdots$ and each $k=1,\cdots, 3^{n}$, let
$$
C_{n,k} = \left \{(x,y)\in \Bbb R^2:
\left (x-\frac{2k-1}{2\cdot 3^n}\right )^2 + y^2
=\frac {1}{4}\cdot 9^{-n} \mbox{and} y\le 0\right \};
$$
let
$$
X = A \cup \left [\bigcup_{n=1}^\infty \left (
\bigcup_{k=1}^{2^{n-1}} B_{n,k}\right )\right ]
\cup
\left [\bigcup_{n=0}^\infty \left ( \bigcup_{k=1}^{3^{n}} C_{n,k}\right
)\right ]
$$
\medskip
1) $X$ is an example of an \g{hlc} \g{continuum} which is
not \g{regular}.
2) $X$ is the union of two \g{regular} \g{continua}.
See Nadler 10.59, p.193.
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\end{rawhtml}
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\addtocounter{figure}{-1}
\begin{latexonly}
\begin{figure}
\centerline
{{\includegraphics{../s/c0140/s0010/e0090/a.eps}}}
\caption[
(
A
)
Semicircle Continuum]{
(
A
)
Semicircle Continuum}
\end{figure}
\end{latexonly}
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\subsection{
Boxes Continuum}
\setcounter{equation}{0}
\h{Boxes Continuum} is a nice \g{continuum}.
Let $A = \{(x,2^{-n+1})\in \Bbb R^2: 0 \le x \le 1\}$;
for each $n=0,1,2,\cdots$ and each $m=0,\cdots, 2^{n+1}$,
let
$$
B_{n,m} = \left \{(m\cdot 2^{-n-1},y)\in \Bbb R^2: 0\le y
\le2^{-n}\right \};
$$
finally, let
$$
X = \left [\bigcup_{n=1}^\infty A_n\right ]
\cup
\left [\bigcup_{n=0}^\infty \left ( \bigcup_{m=0}^{2^{n+1}} B_{n,m}\right
)\right ].
$$
\medskip
1) $X$ is an example of an non-\g{hlc} \g{continuum} which
is not \g{regular}.
2) $X$ is the union of two \g{regular} \g{continua}.
See Nadler 10.38, p.186.
\setcounter{figure}{\value{subsection}}
\addtocounter{figure}{-1}
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\end{rawhtml}
\setcounter{figure}{\value{subsection}}
\addtocounter{figure}{-1}
\begin{latexonly}
\begin{figure}
\centerline
{{\includegraphics{../s/c0140/s0010/e0100/a.eps}}}
\caption[
(
A
)
Union of two dendrites, a non-hlc
Boxes Continuum]{
(
A
)
Union of two dendrites, a non-hlc
Boxes Continuum}
\end{figure}
\end{latexonly}
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\section{
New examples 2}
New examples are nice.
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%%%%%%%%% /c0140/s0020/e0010
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\subsection{
Graph $K_{3,3}$}
\setcounter{equation}{0}
\h{Graph $K_{3,3}$} is a not embedable in $\Bbb R^2$.
\setcounter{figure}{\value{subsection}}
\addtocounter{figure}{-1}
\begin{rawhtml}
\end{rawhtml}
\setcounter{figure}{\value{subsection}}
\addtocounter{figure}{-1}
\begin{latexonly}
\begin{figure}
\centerline
{{\includegraphics{../s/c0140/s0020/e0010/a.eps}}}
\caption[
(
A
)
Graph $K_{3,3}$]{
(
A
)
Graph $K_{3,3}$}
\end{figure}
\end{latexonly}
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\subsection{
Graph $K_5$}
\setcounter{equation}{0}
\h{Graph $K_5$} is a simple \g{continuum}.
\setcounter{figure}{\value{subsection}}
\addtocounter{figure}{-1}
\begin{rawhtml}
\end{rawhtml}
\setcounter{figure}{\value{subsection}}
\addtocounter{figure}{-1}
\begin{latexonly}
\begin{figure}
\centerline
{{\includegraphics{../s/c0140/s0020/e0020/a.eps}}}
\caption[
(
A
)
Graph $K_5$]{
(
A
)
Graph $K_5$}
\end{figure}
\end{latexonly}
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\subsection{
Space Filling Curve}
\setcounter{equation}{0}
\h{Space Filling Curve} is a strange \g{continuum}.
\setcounter{figure}{\value{subsection}}
\addtocounter{figure}{-1}
\begin{rawhtml}
\end{rawhtml}
\setcounter{figure}{\value{subsection}}
\addtocounter{figure}{-1}
\begin{latexonly}
\begin{figure}
\centerline
{{\includegraphics{../s/c0140/s0020/e0030/a.eps}}}
\caption[
(
A
)
An example of
Space Filling Curve]{
(
A
)
An example of
Space Filling Curve}
\end{figure}
\end{latexonly}
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\subsection{
Universal Arc-like Continuum}
\setcounter{equation}{0}
\h{Universal Arc-like Continuum} is a useful \g{continuum}.
\setcounter{figure}{\value{subsection}}
\addtocounter{figure}{-1}
\begin{rawhtml}
\end{rawhtml}
\setcounter{figure}{\value{subsection}}
\addtocounter{figure}{-1}
\begin{latexonly}
\begin{figure}
\centerline
{{\includegraphics{../s/c0140/s0020/e0040/a.eps}}}
\caption[
(
A
)
First steps in the construction of the
Universal Arc-like Continuum]{
(
A
)
First steps in the construction of the
Universal Arc-like Continuum}
\end{figure}
\end{latexonly}
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\subsection{
Lakes of Wada}
\setcounter{equation}{0}
\h{Lakes of Wada} is a strange \g{continuum}.
\setcounter{figure}{\value{subsection}}
\addtocounter{figure}{-1}
\begin{rawhtml}
\end{rawhtml}
\setcounter{figure}{\value{subsection}}
\addtocounter{figure}{-1}
\begin{latexonly}
\begin{figure}
\centerline
{{\includegraphics{../s/c0140/s0020/e0050/a.eps}}}
\caption[
(
A
)
A construction of the
Lakes of Wada]{
(
A
)
A construction of the
Lakes of Wada}
\end{figure}
\end{latexonly}
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\subsection{
Borsuk Fan}
\setcounter{equation}{0}
\h{Borsuk Fan} is a nice \g{continuum}.
\setcounter{figure}{\value{subsection}}
\addtocounter{figure}{-1}
\begin{rawhtml}
\end{rawhtml}
\setcounter{figure}{\value{subsection}}
\addtocounter{figure}{-1}
\begin{latexonly}
\begin{figure}
\centerline
{{\includegraphics{../s/c0140/s0020/e0060/a.eps}}}
\caption[
(
A
)
thick segments are 'above' the thin ones,
Borsuk Fan]{
(
A
)
thick segments are 'above' the thin ones,
Borsuk Fan}
\end{figure}
\end{latexonly}
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\subsection{
Ingram's Atriodic Continuum}
\setcounter{equation}{0}
INGRAM COMMENT (modified):
Figure A represents the first and second composite
of a mapping $f$ of a triod $T = OA \cup OB \cup OC$, onto
itself. The mapping $f$ has the property that $f(O) = B$,
$f(A) = C$, $f(B)= C$, $f(C) = C$, $f(C/2) = O$, $f(B/3)
=O$, $f(B/2) = A/2$, $f(2B/3)= O$, $f(A/4)= O$, $f(A/2) = A$, $f(3A/4) = O$
and the mapping is piecewise
linear. In the picture, the black square represents $O$,
the green arm is the image of $OA$, the blue arm is the
image of $OB$ and the red arm is the image of $OC$. The
inverse limit of the inverse sequence, ${T,f}$, is an
atriodic tree-like continuum which is not chainable [Fund.
Math., 77(1972), 99
- 107].
We call it an \h{Ingram's Atriodic Continuum}.
If one imagines that the fat triod in the top of the Fgure A
represents the first of a sequence of tree chains covering
the inverse limit then the blue-green-red folded triod
represents the next terms of the sequence of covers.
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\caption[
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First steps in the construction of the
Ingram's Atriodic Continuum]{
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First steps in the construction of the
Ingram's Atriodic Continuum}
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\subsection{
Double Warsaw Circle}
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We call it a \h{Double Warsaw Circle}.
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\caption[
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Double Warsaw Circle]{
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Double Warsaw Circle}
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\subsection{
Warsaw Circle}
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We call it a \h{Warsaw Circle}.
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\centerline
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\caption[
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Warsaw Circle]{
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Warsaw Circle}
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\subsection{
Crooked Warsaw Circle}
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We call it a \h{Crooked Warsaw Circle}.
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\caption[
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Crooked Warsaw Circle]{
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Crooked Warsaw Circle}
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\section{
New examples 3}
New examples 3 are useful.
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\subsection{
Young Spiral}
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We call it a \h{Young Spiral}. See Roman Ma\v nka : On
spirals and fixed point property, Fund. Math. 144(1994), p.
1-9.
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\caption[
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Young Spiral]{
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Young Spiral}
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\subsection{
Young Space}
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We call it a \h{Young Space}. See Roman Ma\v nka : On
spirals and fixed point property, Fund. Math. 144(1994), p.
1-9.
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\centerline
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\caption[
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Young Space]{
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Young Space}
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\subsection{
Bing Space}
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We call it a \h{Bing Space}. See Roman Ma\v nka : On spirals
and fixed point property, Fund. Math. 144(1994), p. 1-9.
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\begin{figure}
\centerline
{{\includegraphics{../s/c0140/s0030/e0030/a.eps}}}
\caption[
(
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Bing Space]{
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Bing Space}
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\subsection{
Universal Smooth Dendroid}
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We call it a \h{Universal Smooth Dendroid}.
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\caption[
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Universal Smooth Dendroid]{
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Universal Smooth Dendroid}
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\subsection{
Claytor's Curve A}
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We call it a \h{Claytor's Curve A}.
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\caption[
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Claytor's Curve A]{
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Claytor's Curve A}
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\subsection{
Claytor's Curve B}
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We call it a \h{Claytor's Curve B}.
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\caption[
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Claytor's Curve B]{
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Claytor's Curve B}
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\subsection{
Sin Addiction Continuum}
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We call it a \h{Sin Addiction Continuum}.
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\caption[
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Sin Addiction Continuum]{
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Sin Addiction Continuum}
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\subsection{
Dumbbell}
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We call it a \h{Dumbbell}.
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\caption[
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Dumbbell]{
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\subsection{
Figure Eight}
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We call it a \h{Figure Eight}.
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Figure Eight]{
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Figure Eight}
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\subsection{
Noose}
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We call it a \h{Noose}.
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\caption[
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Noose]{
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Noose}
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\section{
New examples 4}
New examples 4 are useful.
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\subsection{
Theta Curve}
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We call it a \h{Theta Curve}.
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\caption[
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Theta Curve]{
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Theta Curve}
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\subsection{
Trapezoid Basins Continuum}
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We call it a \h{Trapezoid Basins Continuum}.
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\caption[
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Trapezoid Basins Continuum]{
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Trapezoid Basins Continuum}
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\subsection{
Dragon Continuum}
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We call it a \h{Dragon Continuum}. It is a fix point free
continuum.
L. Fearnley, D. G. Wright : Geometric realization of a
Bellamy Continuum, BUll. London Math. Soc. 15(1993),
177-183.
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\end{rawhtml} \setcounter{figure}{\value{subsection}} \addtocounter{figure}{-1} \begin{latexonly} \begin{figure} \centerline {{\includegraphics{../s/c0140/s0040/e0030/a.eps}}} \caption[ ( A ) Dragon Continuum]{ ( A ) Dragon Continuum} \end{figure} \end{latexonly} \setcounter{figure}{\value{subsection}} \addtocounter{figure}{-1} \begin{latexonly} \begin{figure} \centerline {{\includegraphics{../s/c0140/s0040/e0030/b.eps}}} \caption[ ( B ) Dragon Continuum]{ ( B ) Dragon Continuum} \end{figure} \end{latexonly} %%%%%%%%% END %%%%%%%%% %%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% START DEFINITIONS \appendix \chapter{Definitions} %% %% \section*{ absolute end point } A point $p$ of a \g{continuum} $X$ is called an \h{absolute end point} of $X$ if $X \setminus \{p\}$ is a \g{composant} of $X$. \section*{ absolute retract } An \h{absolute retract} is a space $X$ such that if $X$ is embedded as a closed subset $X'$ of a space $Y$, then $X'$ is a \g{retract} of $Y$. \section*{ absolutely terminal continuum } A proper \g{subcontinuum} $K$ of a \g{continuum} $X$ is said to be an \h{absolutely terminal continuum} of $X$ provided that $K$ is a \g{terminal continuum} of each \g{subcontinuum} $L$ of $X$ which properly contains $K$. \section*{ acyclic } A \cm $X$ is said to be \h{acyclic} provided that each mapping from $X$ into the unit circle $\U^1$ is homotopic to a constant mapping, i.e., for all mappings $f: X \to \U^1$ and $c: X \to \{p\} \subset \U^1$ there exists a mapping $h: X \times [0,1] \to \U^1$ 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]{nadler1992}). \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]{fgl1981}). 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]{fgl1981}). 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]{mackowiak1979}): \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]{whyburn1971}). \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]{fgl1981}). \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]{nadler1978}). \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]{nadler1978}, and it is continuous on a dense $G_\delta$ subset of $C(X)$, \cite[Corollary 15.3, p. 515]{nadler1978}. 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]{nadler1978}). \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]{whyburn1971}). \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]{dugundji1966}). \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]{mackowiak1979}). \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]{nadler1978}). 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]{nadler1978}). \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{nadler1992}, as well as in \cite{kuratowski1968}. \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]{nadler1992}. 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]{bing1951}. \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]{mackowiak1979}). \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]{mackowiak1979}): \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]{mackowiak1979}): \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 } A continuous function $f:X \to Y$ is said to be \h{monotone} 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]{np1993}). 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]{mackowiak1979}): \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 } A mapping $f$ of a space $X$ into a space $Y$ is \h{open} 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]{kuratowski1968}. 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]{jjc1962}). \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]{fgl1981}. \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]{nadler1978}). \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]{mackowiak1979}) \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]{fgl1981}). \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 \h{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]{nadler1978}). \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]{mackowiak1979}): \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]{np1993}). \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]{ce1970}). \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]{benett1971}). \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]{fgl1981}) $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]{owens1986}). \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]{mackowiak1979}) \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]{owens1986} 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]{mackowiak1979}) \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]{lum1974}). \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} \gl{$n$-dimensional hereditarily indecomposable continua (R.H.Bing)}{$n$-dimensional hereditarily indecomposable continua (R.H.Bing)} \newline \gl{Bing Space}{Bing Space} \newline \gl{Borsuk Fan}{Borsuk Fan} \newline \gl{Boxes Continuum}{Boxes Continuum} \newline \gl{Buckethandle}{Buckethandle} \newline \gl{Cantor Comb}{Cantor Comb} \newline \gl{Cantor Fan}{Cantor Fan} \newline \gl{Cantor Interaction}{Cantor Interaction} \newline \gl{Cantor Meanders}{Cantor Meanders} \newline \gl{Cantor Snake}{Cantor Snake} \newline \gl{Claytor's Curve A}{Claytor's Curve A} \newline \gl{Claytor's Curve B}{Claytor's Curve B} \newline \gl{Crooked Warsaw Circle}{Crooked Warsaw Circle} \newline \gl{Cruller}{Cruller} \newline \gl{Double Buckethandle}{Double Buckethandle} \newline \gl{Double Topologists's Sin Curve}{Double Topologists's Sin Curve} \newline \gl{Double Warsaw Circle}{Double Warsaw Circle} \newline \gl{Dragon Continuum}{Dragon Continuum} \newline \gl{Dumbbell}{Dumbbell} \newline \gl{Dyadic Selenoid}{Dyadic Selenoid} \newline \gl{Figure Eight}{Figure Eight} \newline \gl{Gehman Dendrite with Ski}{Gehman Dendrite with Ski} \newline \gl{Gehman Dendrite}{Gehman Dendrite} \newline \gl{Graph $K_5$}{Graph $K_5$} \newline \gl{Graph $K_{3,3}$}{Graph $K_{3,3}$} \newline \gl{Harmonic Fan}{Harmonic Fan} \newline \gl{Harmonic Shredded Fan}{Harmonic Shredded Fan} \newline \gl{Hilbert's cube}{Hilbert's cube} \newline \gl{Ingram's Atriodic Continuum}{Ingram's Atriodic Continuum} \newline \gl{Lakes of Wada}{Lakes of Wada} \newline \gl{Lelek Fan}{Lelek Fan} \newline \gl{M-Continuum}{M-Continuum} \newline \gl{Makuchowski Twin}{Makuchowski Twin} \newline \gl{Makuchowski Umbrella}{Makuchowski Umbrella} \newline \gl{Menger's universal curve. Its Anderson's characterization.}{Menger's universal curve. Its Anderson's characterization.} \newline \gl{Noose}{Noose} \newline \gl{Open Sin}{Open Sin} \newline \gl{Pseudo-arc}{Pseudo-arc} \newline \gl{Pseudo-circle}{Pseudo-circle} \newline \gl{Root}{Root} \newline \gl{Semicircle Continuum}{Semicircle Continuum} \newline \gl{Sierpi\'nski Tringle}{Sierpinski Triangle} \newline \gl{Sierpi\'nski Universal Plane Curve}{Sierpinski Universal Plane Curve} \newline \gl{Sin Addiction Continuum}{Sin Addiction Continuum} \newline \gl{Snake}{Snake} \newline \gl{Space Filling Curve}{Space Filling Curve} \newline \gl{The pseudoarc}{The pseudoarc} \newline \gl{The pseudocircle}{The pseudocircle} \newline \gl{Theta Curve}{Theta Curve} \newline \gl{Topologist's Sin Curve}{Topologist's Sin Curve} \newline \gl{Trapezoid Basins Continuum}{Trapezoid Basins Continuum} \newline \gl{Universal Arc-like Continuum}{Universal Arc-like Continuum} \newline \gl{Universal Dendrite $D_3$}{Universal Dendrite $D_3$} \newline \gl{Universal Dendrite $D_4$}{Universal Dendrite $D_4$} \newline \gl{Universal Dendrite $D_6$}{Universal Dendrite $D_6$} \newline \gl{Universal Smooth Dendroid}{Universal Smooth Dendroid} \newline \gl{Wa\.zewski's Universal Dendrite}{Wazewski's Universal Dendrite} \newline \gl{Warsaw Circle}{Warsaw Circle} \newline \gl{Whyburn's Curve}{Whyburn's Curve} \newline \gl{Young Space}{Young Space} \newline \gl{Young Spiral}{Young Spiral} \newline \gl{Zig-zag Circle}{Zig-zag Circle} \newline %%%%%%%%% END LIST OF EXAMPLES %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %% \printindex % % Alphabetization and navigation within the index % ...these special index entries must come *after* the \printindex % else half of the hyperlinks will point to the preceding page. % \begin{htmlonly} \newcommand{\indexAlpha}[5]{\index{#1@\htmlref{_}{#2}% \htmlref{\HTML{SUB}{\LARGE #3}}{AZ}\htmlref{_}{#4}\label{#5}| }} % \indexAlpha{\$}{Z}{\$}{dot}{doll}% \indexAlpha{.}{doll}{~.~}{A}{dot}% \indexAlpha{A}{dot}{A}{B}{A}% \indexAlpha{B}{A}{B}{C}{B}% \indexAlpha{C}{B}{C}{D}{C}% \indexAlpha{D}{C}{D}{E}{D}% \indexAlpha{E}{D}{E}{F}{E}% \indexAlpha{F}{E}{F}{G}{F}% \indexAlpha{G}{F}{G}{H}{G}% \indexAlpha{H}{G}{H}{I}{H}% \indexAlpha{I}{H}{I}{J}{I}% %\indexAlpha{I}{H}{I}{L}{I}% %\indexAlpha{J}{I}{J, K}{L}{K}% \indexAlpha{J}{I}{J}{K}{J}% \indexAlpha{K}{J}{K}{L}{K}% \indexAlpha{L}{K}{L}{M}{L}% \indexAlpha{M}{L}{M}{N}{M}% \indexAlpha{N}{M}{N}{O}{N}% \indexAlpha{O}{N}{O}{P}{O}% \indexAlpha{P}{O}{P}{Q}{P}% %\indexAlpha{P}{O}{P}{R}{P}% %\indexAlpha{Q}{P}{Q}{S}{R}% \indexAlpha{Q}{P}{Q}{R}{Q}% \indexAlpha{R}{Q}{R}{S}{R}% \indexAlpha{S}{R}{S}{T}{S}% \indexAlpha{T}{S}{T}{U}{T}% \indexAlpha{U}{T}{U}{V}{U}% \indexAlpha{V}{U}{V}{W}{V}% \indexAlpha{W}{V}{W}{X}{W}% \indexAlpha{X}{W}{X}{Y}{X}% %\indexAlpha{X}{W}{X}{Z}{X}% %\indexAlpha{Y}{X}{Y, Z}{doll}{Z}% \indexAlpha{Y}{X}{Y}{Z}{Y}% \indexAlpha{Z}{Y}{Z}{doll}{Z}% % % %% This is an alphabetical navigation panel. \index{@\label{AZ}\textbf{\LARGE \htmlref{\$}{doll} \htmlref{.}{dot} \htmlref{A}{A}% \htmlref{B}{B} \htmlref{C}{C} \htmlref{D}{D} \htmlref{E}{E} \htmlref{F}{F} % \htmlref{G}{G} \htmlref{H}{H} \htmlref{I}{I} \htmlref{J}{J} \htmlref{K}{K} % \htmlref{L}{L} \htmlref{M}{M} \htmlref{N}{N} \htmlref{O}{O} \htmlref{P}{P} % \htmlref{Q}{Q} \htmlref{R}{R} \htmlref{S}{S} \htmlref{T}{T} \htmlref{U}{U} % \htmlref{V}{V} \htmlref{W}{W} \htmlref{X}{X} \htmlref{Y}{Y} \htmlref{Z}{Z}}\\ \htmlrule[all]| } \end{htmlonly} %% %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% START REFERENCES \begin{thebibliography}{999} %% %% \newcommand{\citej}{\emph} \newcommand{\citeb}{\rm} \newcommand{\citen}{\textbf} \bibitem{acp20xx} D. 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