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\topmatter
\title
Problem Book (\revised)
\endtitle
\endtopmatter
%\rightheadtext{Problem Book}
%\leftheadtext{Pavel Pyrih}
\vskip 1in
\document



Open Problem Seminar is devoted to the solving of the open
problems from Continuum Theory and close topics from General
Topology and Real Analysis.

(Prohibitted terms : cathegory, measure, dual, hyperspace, span, ...{\tt ;-)}
\medskip
Simply formulated problems are prefered.
\medskip
The Seminar puts special emphasis on active discussing
unsolved problems, both old and new, which appear in its
area of interest, and of course on their SOLVING.
\medskip
Priority is given to the students and their solutions of the
problems.

\medskip
The Seminar has its own "PROBLEM BOOK" circulating during
the meetings among the participants, who are encouraged to
skim it, check, read the notes, and make their comments both
in the "PROBLEM BOOK" and aloud.

\bigskip
\medskip
The Seminar is open to the broad mathametical community and
all visitors are kindly invited to participate the Seminar.
\bigskip
\smallskip

Pavel Pyrih


\medskip
internet: {\tt http://www.karlin.mff.cuni.cz/\~{}pyrih}

e-mail: {\tt pyrih\@{}karlin.mff.cuni.cz}

phone to the office: 00420-2-21913268

phone home: 00420-2-51556298 (from 7:00 to 20:00 GMT)

office:  Sokolovsk\'a 83, floor 2nd, door No. 211, Department
of Mathematical Analysis, Charles University, Prague.

address: Pavel Pyrih, KMA MFF UK, Sokolovsk\'a 83, CZ -- 186 75
Prague 8, Czech Republic.



\vfill\eject


\bigskip
\bigskip
\bigskip

\bf Agreement \rm
\bigskip
\bigskip

{\bf 1.}  Research on topics actively discussed during the Open Problem Seminar
in the common academic year will be published as a joint paper
of active participants of the seminar with a footnote :

\ \newline
"Joint research of the Open Problem Seminar, Charles University, Prague."

\medskip

Research on topics actively discussed during the Open Problem Seminar
in the last academic year can be published by any participant of the seminar
with a footnote :

\ \newline
"Research was discussed in the Open Problem Seminar, Charles  University, Prague."

\medskip

Research on topics actively discussed during the Open Problem Seminar
two years ago can be freely published by any participant of the seminar.


\medskip

\bigskip
\bigskip

{\bf 2.} Published papers with the solutions of presented problems can include the
following information:

\medskip

This problem also appeared in the Problem book of the Open
Problem Seminar held at Department of Mathematical analysis
at Charles University ([2], Problem 28]).

\medskip

[2] P. Pyrih,
{\it Problem book of the Open Problem Seminar},
Charles University, Prague, 2001.

\vfill\eject




\proclaim{\bf Problem 1}
Does every totally non-semi-locally connected continuum
contain a dense $G_{\delta}$ set of weak cut points?
\endproclaim

(E.~E.~Grace, 02/02/72,  Problem 22 of the Houston Problem
Book)
\demo{ Definition}
A  {\bf continuum} is a nonempty, compact, connected metric
space.
\enddemo
\demo{ Definition}
A continuum $M$ is {\bf 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 {\bf totally non-semi-locally connected} if it is
not semi-locally connected at any point.
\enddemo
\demo{ Definition}
A point $x$ of a continuum $M$ is a {\bf weak cut point} of $M$
if there are two points $p$ and $q$ of $M$ such that every
subcontinuum of $M$ that contains both $p$ and $q$ also contains
$x$.
\enddemo

\what

\proclaim{\bf Problem 2}
If $C$ is a connected Borel subset of a finitely Suslinean
continuum $X$, is $C$ arcwise connected?
\endproclaim

(E.~D.~Tymchatyn, 09/26/74,  Problem 66   of the Houston
Problem Book)

\demo{ Definition}
A  {\bf continuum} is a nonempty, compact, connected metric
space.
\enddemo
\demo{ Definition}
An  {\bf arc} is any space which is homeomorphic to the
closed interval $[0,1]$.
\enddemo
\demo{ Definition}
A continuum $X$ is {\bf finitely Suslinean} if, for each positive
number $\eps$, $X$ does not contain infinitely many mutually
exclusive subcontinua of diameter greater than $\eps$.
\enddemo
\demo{ Definition}
The family $\Cal F$ of {\bf Borel sets} of a space $S$ is the
smallest family satisfying the conditions:
\roster
\item  Every closed set belongs to $\Cal F$;
\item  If $X$ is in $\Cal F$, then $S \setminus X$ is in $\Cal
F$; and
\item  The countable intersection of elements of $\Cal F$ belongs
to $\Cal F$.
\endroster
\enddemo
\par\medpagebreak
\noindent Comment:
Yes, for subsets of regular curves. (D. ~H. ~Fremlin, 04/12/90)
\vertspace

\what


\proclaim{\bf Problem 3}
Suppose $X$ is a continuum such that each $\sigma$-connected
subset of $X$ is a semi-continuum.  Is $X$ locally
connected?
\endproclaim

(J.~Grispolakis, 01/27/75,  Problem  72  of the Houston
Problem Book)

\demo{ Definition}
A  {\bf continuum} is a nonempty, compact, connected metric
space.
\enddemo
\demo{ Definition}
A {\bf semi-continuum} is a continuumwise connected point set.
\enddemo
\demo{ Definition}
A  topological space $(X,\tau)$ is said to be {\bf
$\sigma$-connected} provided that $X$ is not the union of
more than one and at most countably many nonempty, mutually
disjoint, closed subsets.
\enddemo
\demo{ Definition}
Let $X$ be a continuum and $A\subset X$. Then, $X$ is said
to be {\bf irreducible about} $A$ provided no proper
subcontinuum of $X$ contains $A$. A continuum $X$ is said to
be  {\bf irreducible} provided that $X$ is irreducinle about
$\{p,q\}$ for some $p,q \in X$.
\enddemo
\demo{ Definition}
A continuum $X$ is said to be  {\bf decomposable} provided
that $X$ can be written as the union of two proper
subcontinua.
\enddemo
\demo{ Definition}
A continuum $X$ is said to be  {\bf hereditarily
decomposable} provided that each nondegenerate subcontinuum
of $X$ (as well as $X$ itself) is decomposable.
\enddemo

\what


\proclaim{\bf Problem 4 }
Suppose $X$ is a continuum such that each $\sigma$-connected
$F_{\sigma}$ subset of $X$ is a semi-continuum.
Is $X$ semi-aposyndetic?
\endproclaim

 (J.~Grispolakis, 01/27/75,  Problem  73  of the Houston Problem
Book)

\demo{ Definition}
A  {\bf continuum} is a nonempty, compact, connected metric
space.
\enddemo
\demo{ Definition}
A  topological space $(X,\tau)$ is said to be {\bf
$\sigma$-connected} provided that $X$ is not the union of
more than one and at most countably many nonempty, mutually
disjoint, closed subsets.
\enddemo
\demo{ Definition}
A space $X$ is {\bf semi-aposyndetic} if, for each two of
its points, $X$ is aposyndetic at one of them with respect
to the other.
\enddemo



\what


\proclaim{\bf Problem 5 }
Is it true that a continuum $X$ is regular if and only if every
infinite sequence of mutually disjoint connected subsets of $X$
is a null sequence?
\endproclaim

(E.~D.~Tymchatyn, 03/03/75,  Problem 77   of the Houston
Problem Book)

\demo{ Definition}
A  {\bf continuum} is a nonempty, compact, connected metric
space.
\enddemo
\demo{ Definition}
A continuum is {\bf regular} if each two of its points can be
separated by a finite point set.
\enddemo


\what

\proclaim{\bf Problem 6}
Is it true that no biconnected set with a dispersion point can be
embedded into a rational continuum?
\endproclaim

(J.~Grispolakis, 03/03/75,  Problem  79   of the Houston
Problem Book)

\demo{ Definition}
A  {\bf continuum} is a nonempty, compact, connected metric
space.
\enddemo
\demo{ Definition}
A point $p$ of a nondegenerate connected space $X$ is an {\bf
explosion point}, or {\bf dispersion point}, of $X$ provided that
$X \setminus \{p\}$ is totally disconnected.
\enddemo
\demo{ Definition}
A space is {\bf biconnected} if it is not the sum of two
mutually exclusive nondegenerate connected point sets.
\enddemo
\demo{ Definition}
A continuum is {\bf rational} if each two of its points can be
separated by a countable point set.
\enddemo

\what

\proclaim{\bf Problem 7}
Is the confluent image of an arc-like continuum arc-like?
\endproclaim

(A.~Lelek, 10/21/75,  Problem  84   of the Houston Problem
Book)

\demo{ Definition}
A  {\bf continuum} is a nonempty, compact, connected metric
space.
\enddemo
\demo{ Definition}
An  {\bf arc} is any space which is homeomorphic to the
closed interval $[0,1]$.
\enddemo
\demo{ Definition}
A mapping $f$ of a compact space $X$ onto a compact space $Y$ is
{\bf confluent} if, for each continuum $C$ in $Y$, each component
of $f^{-1}(C)$ is mapped onto $C$ by $f$.
\enddemo
\demo{ Definition}
A mapping $f$ from a space $X$ onto a space $Y$ is {\bf locally
confluent} if, for each point $y$ of $Y$, there is an open set
$U$ containing $y$ such that $f^{-1}(\, \Cl{U} \,)$ is confluent.
\enddemo
\demo{ Definition}
If $X$ is a metric space, a mapping $f$ from $X$ to a space
$Y$ is an {\bf $\eps$-map} if, for each point $y$ of
$Y$, $\diam\(f^{-1}\(y\)\) \leq \eps$. If $\Cal C$ is a
collection of continua, a continuum $M$ is {\bf $\Cal
C$-like} if, for every positive number $\eps$, there
exists an $\eps$-map of $M$ onto an element of $\Cal C$.
In particular, a continuum is {\bf tree-like} if, for some
collection $\Cal C$ of trees, $M$ is $\Cal C$-like.
\enddemo






\what


\proclaim{\bf Problem 8 }
Is there an hereditarily equivalent continuum other than an arc
or a pseudo-arc?
\endproclaim

(H.~Cook, 11/02/77,  Problem  90  of the Houston Problem
Book)

\demo{ Definition}
A  {\bf continuum} is a nonempty, compact, connected metric
space.
\enddemo
\demo{ Definition}
A continuum is {\bf hereditarily equivalent} if it is
homeomorphic to each of its nondegenerate subcontinua.
\vertspace\enddemo
\demo{ Definition}
An  {\bf arc} is any space which is homeomorphic to the
closed interval $[0,1]$.
\enddemo
\demo{ Definition}
A {\bf pseudo-arc} is the only (up to homeomorphism)
hereditarily indecomposable arc-like continuum.
\enddemo


\what


\proclaim{\bf Problem 9 }
Is the image under a local homeomorphism of a $\delta$-connected
continuum also $\delta$-connected?
\vertspace\endproclaim


 (C.~L.~Hagopian, 04/12/78,  Problem 96   of the Houston Problem
Book)

\demo{ Definition}
A  {\bf continuum} is a nonempty, compact, connected metric
space.
\enddemo
\demo{ Definition}
A continuum is {\bf $\delta$-connected} if each two of its
points can be connected by an hereditarily decomposable
irreducible subcontinuum.
\enddemo
\demo{ Definition}
Let $X$ be a continuum and $A\subset X$. Then, $X$ is said
to be {\bf irreducible about} $A$ provided no proper
subcontinuum of $X$ contains $A$. A continuum $X$ is said to
be  {\bf irreducible} provided that $X$ is irreducinle about
$\{p,q\}$ for some $p,q \in X$.
\enddemo
\demo{ Definition}
A continuum $X$ is said to be  {\bf decomposable} provided
that $X$ can be written as the union of two proper
subcontinua.
\enddemo
\demo{ Definition}
A continuum $X$ is said to be  {\bf hereditarily
decomposable} provided that each nondegenerate subcontinuum
of $X$ (as well as $X$ itself) is decomposable.
\enddemo

\what

\proclaim{\bf Problem 10 }
Can it be proven, without extraordinary logical assumptions, that
the plane is not the sum of fewer than $\frak c$ mutually
exclusive continua?
\endproclaim

(H.~Cook, 04/19/78,  Problem 97   of the Houston Problem
Book)

\demo{ Definition}
A  {\bf continuum} is a nonempty, compact, connected metric
space.
\enddemo
\par\medpagebreak
\noindent Comment:
The plane is not the sum of fewer than $\frak c$ mutually
exclusive continua each of which is either Suslinean or locally
connected.  (H.~Cook, 04/19/78)
\vertspace


\what


\proclaim{\bf Problem 11 }
Suppose $M$ is a continuum and $f$ is a monotone mapping of $M$
onto $[0,1]$ such that, if $D$ is a closed proper subset of $M$
which is mapped onto $[0,1]$ by $f$, then $f|D$ is not monotone.
Is $M$ irreducible?
\endproclaim

(L.~Mohler and L.~E.~Ward, 10/18/78,  Problem  101  of the
Houston Problem Book)

\demo{ Definition}
A  {\bf continuum} is a nonempty, compact, connected metric
space.
\enddemo
\demo{ Definition}
A continuous function $f:X \to Y$ is said to be  {\bf
monotone} provided  $f^{-1}(y)$ is connected for each $y \in
Y$.
\enddemo
\demo{ Definition}
Let $X$ be a continuum and $A\subset X$. Then, $X$ is said
to be {\bf irreducible about} $A$ provided no proper
subcontinuum of $X$ contains $A$. A continuum $X$ is said to
be  {\bf irreducible} provided that $X$ is irreducinle about
$\{p,q\}$ for some $p,q \in X$.
\enddemo



\what

\proclaim{\bf Problem 12}
Is the open image of a circle-like continuum always circle-like
or arc-like?
\endproclaim

(J.~B.~Fugate, 10/18/78,  Problem  104   of the Houston
Problem Book)

\demo{ Definition}
A  {\bf continuum} is a nonempty, compact, connected metric
space.
\enddemo
\demo{ Definition}
A mapping $f$ of a space $X$ into a space $Y$ is {\bf open}
if, for each open $A$ in $X$ the image $f(A)$ is open in
$Y$.
\enddemo
\demo{ Definition}
If $X$ is a metric space, a mapping $f$ from $X$ to a space $Y$
is an {\bf $\eps$-map} if, for each point $y$ of $Y$,
$\diam\(f^{-1}\(y\)\) \leq \eps$.
If $\Cal C$ is a collection of continua, a continuum $M$ is
{\bf $\Cal C$-like} if, for every positive number $\eps$,
there exists an $\eps$-map of $M$ onto an element of
$\Cal C$.  In particular, a continuum is {\bf tree-like} if, for
some collection $\Cal C$ of trees, $M$ is $\Cal C$-like.
\enddemo
\demo{ Definition}
A  {\bf circle} is any space which is homeomorphic to the
unit circle.
\enddemo
\demo{ Definition}
An  {\bf arc} is any space which is homeomorphic to the
closed interval $[0,1]$.
\enddemo

\what

\proclaim{\bf Problem 13 }
If $M$ is tree-like and every proper subcontinuum of $M$ is
chainable, is $M$ almost chainable?
\endproclaim

(J.~B.~Fugate, 10/18/78,  Problem 106   of the Houston
Problem Book)

\demo{ Definition}
A  {\bf continuum} is a nonempty, compact, connected metric
space.
\enddemo
\demo{ Definition}
If $X$ is a metric space, a mapping $f$ from $X$ to a space $Y$
is an {\bf $\eps$-map} if, for each point $y$ of $Y$,
$\diam\(f^{-1}\(y\)\) \leq \eps$.
If $\Cal C$ is a collection of continua, a continuum $M$ is
{\bf $\Cal C$-like} if, for every positive number $\eps$,
there exists an $\eps$-map of $M$ onto an element of
$\Cal C$.  In particular, a continuum is {\bf tree-like} if, for
some collection $\Cal C$ of trees, $M$ is $\Cal C$-like.
\enddemo
\demo{ Definition}
A  {\bf tree} is a graph which contains no simple closed
curve.
\enddemo
\demo{ Definition}
A {\bf graph} is a continuum which can be written as the
union of finely many arcs any two of which are either
disjoint or intersect only in one or both their endpoints.
\enddemo
\demo{ Definition}
An  {\bf arc} is any space which is homeomorphic to the
closed interval $[0,1]$.
\enddemo
\demo{ Definition}
A {\bf simple chain} is a finite sequence $L_1, L_2, \dots,L_n$
of open sets such that $L_i$ intersects $L_j$ if and only if
$|i-j| \leq 1$.  The terms of the sequence $L_1, L_2, \dots,L_n$
are called the {\bf links} of the chain.
An {\bf $\eps$-chain} is a chain each of whose links has
diameter less than $\eps$.  A continuum $M$ is {\bf
chainable} if, for each positive number $\eps$, $M$ can be
covered by an $\eps$-chain.
\enddemo
\demo{ Definition}
A continuum $M$ is {\bf almost chainable} if, for each positive
number $\eps$, there exists an open cover $D$ of $M$ with
mesh $\eps$ 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 $\eps$ from some link of $C$.
\enddemo


\what

\proclaim{\bf Problem 14 }
Suppose $M_1, M_2, \dots$ is a sequence of mutually disjoint
continua in the plane
converging to the continuum $M$ homeomorphically.  Is $M$
circle-like or chainable?
\endproclaim

(J.~B.~Fugate, 10/18/78,  Problem  107  of the Houston
Problem Book)

\demo{ Definition}
A  {\bf continuum} is a nonempty, compact, connected metric
space.
\enddemo
\demo {Definition}
The statement that the sequence $M_1, M_2, \dots$ {\bf converges
homeomorphically to the continuum $M$} means there exists a
sequence $h_1, h_2, \dots$ of homeomorphisms such that, for
each positive integer $i$, $h_i$ is a homeomorphism from $M_i$
onto $M$ and for each positive number $\eps$ there exists a
positive integer $N$ such that if $j > N$ then, for all $x$,
dist$(h_j(x),x) < \eps$.
\enddemo
\demo{ Definition}
If $X$ is a metric space, a mapping $f$ from $X$ to a space $Y$
is an {\bf $\eps$-map} if, for each point $y$ of $Y$,
$\diam\(f^{-1}\(y\)\) \leq \eps$.
If $\Cal C$ is a collection of continua, a continuum $M$ is
{\bf $\Cal C$-like} if, for every positive number $\eps$,
there exists an $\eps$-map of $M$ onto an element of
$\Cal C$.  In particular, a continuum is {\bf tree-like} if, for
some collection $\Cal C$ of trees, $M$ is $\Cal C$-like.
\enddemo
\demo{ Definition}
A  {\bf circle} is any space which is homeomorphic to the
unit circle.
\enddemo
\demo{ Definition}
A {\bf simple chain} is a finite sequence $L_1, L_2, \dots,L_n$
of open sets such that $L_i$ intersects $L_j$ if and only if
$|i-j| \leq 1$.  The terms of the sequence $L_1, L_2, \dots,L_n$
are called the {\bf links} of the chain.
An {\bf $\eps$-chain} is a chain each of whose links has
diameter less than $\eps$.  A continuum $M$ is {\bf
chainable} if, for each positive number $\eps$, $M$ can be
covered by an $\eps$-chain.
\enddemo

\what

\proclaim{\bf Problem 15 }
Suppose $M$ is a continuum which is not the sum of a countable
monotonic collection of proper subcontinua.  Is $M$ irreducible
about some finite set?
\endproclaim

(J.~B.~Fugate, 10/18/78,  Problem 113   of the Houston
Problem Book)

\demo{ Definition}
A  {\bf continuum} is a nonempty, compact, connected metric
space.
\enddemo
\demo{ Definition}
Let $X$ be a continuum and $A\subset X$. Then, $X$ is said
to be {\bf irreducible about} $A$ provided no proper
subcontinuum of $X$ contains $A$.
\enddemo
\what

\proclaim{\bf Problem 16 }
Suppose $Y$ is a nondegenerate locally connected continuum and
each cyclic element of $Y$ is a completely regular continuum.  Is
it true that there exists a continuum $X$ in the plane and a
monotone open map of $X$ onto $Y$ such that $f^{-1}(y)$ is a
(nondegenerate) decomposable continuum for each $y$ in $Y$?

\endproclaim

(J.~Krasinkiewicz, 03/07/79,  Problem 119   of the Houston
Problem Book)

\demo{ Definition}
A  {\bf continuum} is a nonempty, compact, connected metric
space.
\enddemo
\demo{ Definition}
A continuum is {\bf completely regular} if each nondegenerate
subcontinuum has a nonempty interior.
\enddemo
\demo{ Definition}
If $M$ is a semi-locally connected continuum, a {\bf cyclic
element} of $M$ is either a cut point of $M$, an end point
of $M$, or a nondegenerate subset of $M$ which is maximal
with respect to being a connected subset of $M$ without cut
points.
\enddemo
\demo{ Definition}
Let $S,T)$ be a connected topological space,
and let $p \in S$. If $S \setminus \{p\}$ is not connected,
then $p$ is called a {\bf cut point of} $S$.
\enddemo
\demo{ Definition}
An {\bf MO-mapping} is a composition $gf$ where $g$ is monotone
and $f$ is open.
\enddemo
\demo{ Definition}
A continuous function $f:X \to Y$ is said to be  {\bf
monotone} provided  $f^{-1}(y)$ is connected for each $y \in
Y$.
\enddemo
\demo{ Definition}
A mapping $f$ of a space $X$ into a space $Y$ is {\bf open}
if, for each open $A$ in $X$ the image $f(A)$ is open in
$Y$.
\enddemo

\par\medpagebreak
\noindent Comment:
The answer is yes if $Y$ itself is a completely regular continuum
in the plane.  (\Mackowiak\ and E.~D.~Tymchatyn, 01/25/81)
\vertspace


\what


\proclaim{\bf Problem 17}
Does there exist a widely connected complete metric space?
\endproclaim

(H.~Cook, 03/16/79,  Problem   123  of the Houston Problem
Book)

\demo{ Definition}
A nondegenerate connected space $X$ is {\bf widely
connected} if each nondegenerate connected subset of $X$ is
dense in $X$.
\enddemo


\what

\proclaim{\bf Problem 18}
Suppose $M$ is a continuum such that, if $G$ is an uncountable
collection of nondegenerate subcontinua of $M$, then some two
elements of $G$ have a nondegenerate continuum in their
intersection.  Does $M$ contain a countable point set that
intersects every nondegenerate subcontinuum of $M$?
\endproclaim

(H.~Cook, 03/26/79,  Problem  131   of the Houston Problem
Book)

\demo{ Definition}
A  {\bf continuum} is a nonempty, compact, connected metric
space.
\enddemo

\what

\proclaim{\bf Problem 19}
Suppose $M$ is an hereditarily indecomposable simple triod-like
continuum such that every proper subcontinuum of $M$ is a
pseudo-arc.  Can $M$ be embedded in the plane?
\endproclaim

(C.~E.~Burgess, 03/27/79,  Problem   135  of the Houston
Problem Book)

\demo{ Definition}
A  {\bf continuum} is a nonempty, compact, connected metric
space.
\enddemo
\demo{ Definition}
A continuum $X$ is said to be  {\bf indecomposable} provided
that $X$ cannot be written as the union of two proper
subcontinua.
\enddemo
\demo{ Definition}
A continuum $X$ is said to be  {\bf hereditarily
indecomposable} provided that each
of its subcontinua is indecomposable.
\enddemo
\demo{ Definition}
The union of three arcs emanating from a point v is called a
{\bf simple triod} provided the singleton {v} is the
intersection of any two of the arcs ; the point v is then
called the {\bf top} of the simple triod, and the arcs are
called its {\bf arms}.
\enddemo
\demo{ Definition}
If $X$ is a metric space, a mapping $f$ from $X$ to a space $Y$
is an {\bf $\eps$-map} if, for each point $y$ of $Y$,
$\diam\(f^{-1}\(y\)\) \leq \eps$.
If $\Cal C$ is a collection of continua, a continuum $M$ is
{\bf $\Cal C$-like} if, for every positive number $\eps$,
there exists an $\eps$-map of $M$ onto an element of
$\Cal C$.  In particular, a continuum is {\bf tree-like} if, for
some collection $\Cal C$ of trees, $M$ is $\Cal C$-like.
\enddemo
\demo{ Definition}
A {\bf pseudo-arc} is the only (up to homeomorphism)
hereditarily indecomposable arc-like continuum.
\enddemo
\demo{ Definition}
An  {\bf arc} is any space which is homeomorphic to the
closed interval $[0,1]$.
\enddemo





\what

\proclaim{\bf Problem 20}
Is there a monotonely refinable map from a regular curve of
finite order onto a topologically different regular curve of
finite order?
\endproclaim

(E.~E.~Grace, 04/11/79,  Problem   137  of the Houston
Problem Book)

\demo{ Definition}
A  {\bf continuum} is a nonempty, compact, connected metric
space.
\enddemo
\demo{ Definition}
A map $r$ from a compact metric space $X$ onto a compact metric
space $Y$ is {\bf (monotonely) refinable} if, for each positive
number $\eps$, there is a (monotone) $\eps$-map $f$ from
$X$ onto $Y$ such that, for each $x$ in $X$,
$d\( f\(x\),r\(x\) \) < \eps$.
\enddemo
\demo{ Definition}
A continuum is {\bf regular} if each two of its points can be
separated by a finite point set.
\enddemo
\demo{ Definition}
A  {\bf curve} is any one-dimensional continuum.
\enddemo
\demo{ Definition}
Let $(X,T)$ be a topological space, and let $A \subset X$.
Let $\beta$ be a cardinal number.
We say that $A$  {\bf 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 $A$ {\bf 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 $\alpha < \beta$.
\enddemo

\what


\proclaim{\bf Problem 21}
Let $\Cal F$ be a class of mappings such that all homeomorphism
are in $\Cal F$ and the composition of any two functions in
$\Cal F$ is also in $\Cal F$.  If $X$ is homogeneous with respect
to $\Cal F$, is there a continuum which is $\Cal F$-equivalent to
$X$ and which is homogeneous?
\endproclaim

 (H.~Cook, 04/25/79,  Problem   150  of the Houston Problem
Book)

\demo{ Definition}
A  {\bf continuum} is a nonempty, compact, connected metric
space.
\enddemo
\demo{ Definition}
A continuum $X$ is {\bf homogeneous} with respect to $\Cal F$ if,
for each two points $a$ and $b$ of $X$, there is a mapping $f$ in
$\Cal F$ of $X$ onto $X$ such that $f(a)=b$.
\enddemo
\demo{ Definition}
If $X$ and $Y$ are continua, we say that $Y$ is
{\bf $\Cal F$-equivalent} to $X$ provided there is a mapping in
$\Cal F$ from $X$ onto $Y$ and a mapping in $\Cal F$ from $Y$
onto $X$.
\enddemo
\par\medpagebreak
\noindent Comment:
The answer is no for the class $\Cal F$ consisting of all
homeomorphisms and all mappings whose range is not homogeneous.
(David Jones, 05/09/79)
\par\medpagebreak
\noindent Comment:
What is the answer if $\Cal F$ is the class of all mappings?  All
monotone mappings?  All open mappings?  All finite to one
mappings?  All confluent mappings? (H.~Cook, 5/11/79)
\par\medpagebreak
\noindent Comment:
No, if $\Cal F$ is the class of all confluent mappings.
(H.~Kato, 08/09/84)
\par\medpagebreak
\noindent Comment:No, if $\Cal F$ is the class of all mappings; take $X$ to be a
harmonic fan as in the paper of P.~Krupski in {\sl Houston J\.
Math\.\/} {\bf 5} (1979), 345--356.  (J.~Prajs, 03/01/89)
\vertspace

\what

\proclaim{\bf Problem 22}
Suppose, in the previous problem, $X$ is a homogeneous continuum,
$Y$ is a continuum which is $\Cal F$-equivalent to $X$, and
$\eps$ is a positive number.  Does there exist a positive
number $\delta$ such that if $a$ and $b$ are two points of $Y$
such that the distance from $a$ to $b$ is less than $\delta$,
then there is a mapping $f$ in $\Cal F$ from $Y$ onto $Y$ such
that $f(a)=b$ and no point of $Y$ is moved a distance more than
$\eps$?
\endproclaim

(H.~Cook, 04/25/79,  Problem  151   of the Houston Problem
Book)

\demo{ Definition}
A  {\bf continuum} is a nonempty, compact, connected metric
space.
\enddemo
\par\medpagebreak
\noindent Comment:
If the answer to problem 150 is `yes', then the answer to this
problem is `no'.  (H.~Cook, 04/25/79)
\vertspace


\what

\proclaim{\bf Problem 23 }
Are planar dendroids (arcwise connected continua) weakly
chainable?
\endproclaim

(J.~Krasinkiewicz, 11/14/79,  Problem 155   of the Houston
Problem Book)

\demo{ Definition}
A  {\bf continuum} is a nonempty, compact, connected metric
space.
\enddemo
\demo{ Definition}
A {\bf dendroid} is an arcwise connected, hereditarily unicoherent continuum.
\enddemo
\demo{ Definition}
An  {\bf arc} is any space which is homeomorphic to the
closed interval $[0,1]$.
\enddemo
\demo{ Definition}
A connected topological space is said to be
{\bf hereditarily unicoherent} provided that
each of its closed, connected subsets is unicoherent.
\enddemo
\demo{ Definition}
A connected topological space $S$ is said to be
{\bf 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.
\enddemo
\demo{ Definition}
A continuum is {\bf weakly chainable} if it is a continuous image
of a chainable continuum.
\enddemo
\demo{ Definition}
A {\bf simple chain} is a finite sequence $L_1, L_2, \dots,L_n$
of open sets such that $L_i$ intersects $L_j$ if and only if
$|i-j| \leq 1$.  The terms of the sequence $L_1, L_2, \dots,L_n$
are called the {\bf links} of the chain.
An {\bf $\eps$-chain} is a chain each of whose links has
diameter less than $\eps$.  A continuum $M$ is {\bf
chainable} if, for each positive number $\eps$, $M$ can be
covered by an $\eps$-chain.
\enddemo




\what

\proclaim{\bf Problem 24}
Let $X$ be a nondegenerate homogeneous hereditarily decomposable
continuum.  Is it true that $X$ is homeomorphic to a circle?
\endproclaim

(J.~Krasinkiewicz, 11/14/79,  Problem  156   of the Houston
Problem Book)

\demo{ Definition}
A  {\bf continuum} is a nonempty, compact, connected metric
space.
\enddemo
\demo{ Definition}
A continuum $X$ is {\bf homogeneous} with respect to $\Cal F$ if,
for each two points $a$ and $b$ of $X$, there is a mapping $f$ in
$\Cal F$ of $X$ onto $X$ such that $f(a)=b$.
\enddemo
\demo{ Definition}
A continuum $X$ is said to be  {\bf hereditarily
decomposable} provided that each nondegenerate subcontinuum
of $X$ (as well as $X$ itself) is decomposable.
\enddemo
\demo{ Definition}
A continuum $X$ is said to be  {\bf decomposable} provided
that $X$ can be written as the union of two proper
subcontinua.
\enddemo
\demo{ Definition}
A  {\bf circle} is any space which is homeomorphic to the
unit circle.
\enddemo


\what

\proclaim{\bf Problem 25 }
Does there exist a thin space with infinitely many points?
\endproclaim

(P.~H.~Doyle, 02/18/80,  Problem  168  of the Houston
Problem Book)

\demo{ Definition}
A topological space $S$ is {\bf thin} if and only if for each two
homeomorphic subsets $A$ and $B$ of $S$, there is a homeomorphism
$h$ of $S$ onto itself such that $h(A)=B$.
\enddemo
\par\medpagebreak
\noindent Comment:
The only known examples of thin spaces are finite spaces which
are the product of a discrete space with an indiscrete space
(i.e. a space $S$ whose only open sets are $S$ and the empty
set).
\vertspace


\what


\proclaim{\bf Problem 26 } Is every weakly chainable atriodic tree-like continuum
chainable?

\endproclaim

(Lee Mohler, 04/16/81,  Problem  171  of the Houston Problem
Book)

\demo{ Definition}
A  {\bf continuum} is a nonempty, compact, connected metric
space.
\enddemo
\demo{ Definition}
A continuum is {\bf weakly chainable} if it is a continuous image
of a chainable continuum.
\enddemo
\demo{ Definition}
A continuum that contains no subcontinuum which is itself a
triod is said to be {\bf atriodic}
\enddemo
\demo{ Definition}
A continuum is called a {\bf triod} provided it contains a
subcontinuum whose complement is the union of three nonempty
pairwise disjoint open sets.
\enddemo
\demo{ Definition}
A  {\bf tree} is a graph which contains no simple closed
curve.
\enddemo
\demo{ Definition}
If $X$ is a metric space, a mapping $f$ from $X$ to a space $Y$
is an {\bf $\eps$-map} if, for each point $y$ of $Y$,
$\diam\(f^{-1}\(y\)\) \leq \eps$.
If $\Cal C$ is a collection of continua, a continuum $M$ is
{\bf $\Cal C$-like} if, for every positive number $\eps$,
there exists an $\eps$-map of $M$ onto an element of
$\Cal C$.  In particular, a continuum is {\bf tree-like} if, for
some collection $\Cal C$ of trees, $M$ is $\Cal C$-like.
\enddemo
\demo{ Definition}
A {\bf graph} is a continuum which can be written as the
union of finely many arcs any two of which are either
disjoint or intersect only in one or both their endpoints.
\enddemo
\demo{ Definition}
An  {\bf arc} is any space which is homeomorphic to the
closed interval $[0,1]$.
\enddemo
\demo{ Definition}
A  {\bf simple closed curve} is any space which is homeomorphic to the
unit circle.
\enddemo
\demo{ Definition}
A {\bf simple chain} is a finite sequence $L_1, L_2, \dots,L_n$
of open sets such that $L_i$ intersects $L_j$ if and only if
$|i-j| \leq 1$.  The terms of the sequence $L_1, L_2, \dots,L_n$
are called the {\bf links} of the chain.
An {\bf $\eps$-chain} is a chain each of whose links has
diameter less than $\eps$.  A continuum $M$ is {\bf
chainable} if, for each positive number $\eps$, $M$ can be
covered by an $\eps$-chain.
\enddemo

\what

\proclaim{\bf Problem 27 }
Does there exist a continuum $M$ such that no monotone continuous
image of $M$ contains a chainable continuum?
\endproclaim

(H.~Cook, 05/13/81,  Problem  172  of the Houston Problem
Book)

\demo{ Definition}
A  {\bf continuum} is a nonempty, compact, connected metric
space.
\enddemo
\demo{ Definition}
A continuous function $f:X \to Y$ is said to be  {\bf
monotone} provided  $f^{-1}(y)$ is connected for each $y \in
Y$.
\enddemo
\demo{ Definition}
A {\bf simple chain} is a finite sequence $L_1, L_2, \dots,L_n$
of open sets such that $L_i$ intersects $L_j$ if and only if
$|i-j| \leq 1$.  The terms of the sequence $L_1, L_2, \dots,L_n$
are called the {\bf links} of the chain.
An {\bf $\eps$-chain} is a chain each of whose links has
diameter less than $\eps$.  A continuum $M$ is {\bf
chainable} if, for each positive number $\eps$, $M$ can be
covered by an $\eps$-chain.
\enddemo






\what

\proclaim{\bf Problem 28}
In a topological space $X$, each set $A$ has its derived set,
$$
A^d = \bigl\{\,x \in X \bigm| x \in \Cl{ A-\{x\} } \,\bigr\}.
$$
Does there exist a $T_0$-space $X$ and a subset $A$ of $X$ such
that the second derived set $(A^d)^d$ is not closed in $X$?

\endproclaim

(A.~Lelek, 05/15/81,  Problem  174   of the Houston Problem
Book)

\medskip

SOLVED!
(answer NO, see O.F.K. Kalenda:
On double-derived sets in topological spaces,
Houston Journal of Math. 26,4(2000), 807--809.)

\medskip


\what

\proclaim{\bf Problem 29 }
Is there a tree-like continuum $X$ on the plane such that $X$ is
weakly chainable but a subcontinuum of $X$ is not?
\endproclaim

(P.~Minc, 01/24/83,  Problem  179  of the Houston Problem
Book)

\demo{ Definition}
A  {\bf continuum} is a nonempty, compact, connected metric
space.
\enddemo
\demo{ Definition}
If $X$ is a metric space, a mapping $f$ from $X$ to a space $Y$
is an {\bf $\eps$-map} if, for each point $y$ of $Y$,
$\diam\(f^{-1}\(y\)\) \leq \eps$.
If $\Cal C$ is a collection of continua, a continuum $M$ is
{\bf $\Cal C$-like} if, for every positive number $\eps$,
there exists an $\eps$-map of $M$ onto an element of
$\Cal C$.  In particular, a continuum is {\bf tree-like} if, for
some collection $\Cal C$ of trees, $M$ is $\Cal C$-like.
\vertspace\enddemo
\demo{ Definition}
A  {\bf tree} is a graph which contains no simple closed
curve.
\enddemo
\demo{ Definition}
A {\bf graph} is a continuum which can be written as the
union of finely many arcs any two of which are either
disjoint or intersect only in one or both their endpoints.
\enddemo
\demo{ Definition}
An  {\bf arc} is any space which is homeomorphic to the
closed interval $[0,1]$.
\enddemo
\demo{ Definition}
A  {\bf simple closed curve} is any space which is homeomorphic to the
unit circle.
\enddemo
\demo{ Definition}
A continuum is {\bf weakly chainable} if it is a continuous image
of a chainable continuum.
\enddemo
\demo{ Definition}
A {\bf simple chain} is a finite sequence $L_1, L_2, \dots,L_n$
of open sets such that $L_i$ intersects $L_j$ if and only if
$|i-j| \leq 1$.  The terms of the sequence $L_1, L_2, \dots,L_n$
are called the {\bf links} of the chain.
An {\bf $\eps$-chain} is a chain each of whose links has
diameter less than $\eps$.  A continuum $M$ is {\bf
chainable} if, for each positive number $\eps$, $M$ can be
covered by an $\eps$-chain.
\enddemo




\what


\proclaim{\bf Problem 30 }
Suppose $X$ is an arcwise connected continuum with a free arc and
with the fixed set property for monotone onto maps.  Is $X$ a
simple closed curve?
\endproclaim

(Y.~Ohsuda, 10/31/83,  Problem  181  of the Houston Problem
Book)

\demo{ Definition}
A  {\bf continuum} is a nonempty, compact, connected metric
space.
\enddemo
\demo{ Definition}
An  {\bf arc} is any space which is homeomorphic to the
closed interval $[0,1]$.
\enddemo
\demo{ Definition}
A {\bf free arc} in $X$ is any subset of $X$ of the  form
$A \setminus \{x,y\}$ where $A$ is an arc in $X$ with end points $x$
and $y$ and $A \setminus \{x,y\}$ is open in $X$.
\enddemo
\demo{ Definition}
A topological space $X$ is said to have the {\bf fixed set
property} for a certain class $\Cal C$ of maps of $X$ onto itself
provided there exists, for each non-empty closed set $A$ in $X$,
a map $f$ in $\Cal C$ such that $f(x) = x$ if and only if $x$ is
in $A$.
\enddemo
\demo{ Definition}
A continuous function $f:X \to Y$ is said to be  {\bf
monotone} provided  $f^{-1}(y)$ is connected for each $y \in
Y$.
\enddemo
\demo{ Definition}
A  {\bf simple closed curve} is any space which is homeomorphic to the
unit circle.
\enddemo

\what


\proclaim{\bf Problem 31 }
Does there exist a continuous function $f$ defined on the closed
unit interval with values in a metric space such that $f$ is at
most $n$-to-one, for some positive integer $n$, and $f$ is not
finitely linear?
\endproclaim

 (A.~Lelek, 05/15/89,  Problem  215  of the Houston Problem
Book)

\demo{ Definition}
A  {\bf continuum} is a nonempty, compact, connected metric
space.
\enddemo
\demo{ Definition}
A function $f\: I \to Y$, where $I$ is a closed interval of the
real line, is called {\bf finitely linear} provided there exists
a positive integer $m$ such that $I$ can be decomposed, for each
$\eps > 0$, into a finite number of closed subintervals
$I_1, I_2, \dots, I_k$ each of length less than $\eps$ and
with the property that the set $f(I_i)$ meets at most $m$ of the
sets $f(I_1), f(I_2), \dots, f(I_k)$ for
$i = 1, 2, \dots, k$.\enddemo
\demo{ Comment}
The usefulness of finitely linear functions has been shown
in {\sl Fund\. Math\.\/} {\bf 55} (1964), pp\. 199--214.
(A.~Lelek, 05/15/89)
\enddemo

\what


\proclaim{\bf Problem 32 }
Does there exist a weakly
Suslinian, locally connected continuum, which can be
written as an uncountable disjoint union of proper
subcontinua only finitely many of which are
degenerate?
\endproclaim

 (P.~Pyrih,  to appear in IASI)

\demo{ Definition}
A  {\bf continuum} is a nonempty, compact, connected metric
space.
\enddemo
\demo{ Definition}
A continuum $X$ is said to
be {\bf weakly Suslinian} provided that $X$ is not
the union of more than one pairwise disjoint
nondegenerate subcontinua of $X$.
\enddemo


\what


\proclaim{\bf Problem 33 }
Let a hereditarily unicoherent continuum $X$
have the property that each its proper subcontinuum
is HU-terminal if and only if it is absolutely terminal.
Must then $X$ be atriodic?
\endproclaim

 (J. J. Charatonik, Absolutely terminal continua
 and confluent mappings, Comment. Math. Univ. Carolinae 32,2
 (1991), p. 381)

\medskip

SOLVED!
(answer NO, see
P. Pyrih, T. B\'arta, D. Op\v{e}la, R. \v{S}\'amal:
A continuum where absolute terminality and HU-terminality coincide,
Questions and Answers in General Topology 17,2(1999), 233--235.)

\medskip

\demo{ Definition}
A  {\bf continuum} is a nonempty, compact, connected metric
space.
\enddemo
\demo{ Definition}
A connected topological space is said to be
{\bf hereditarily unicoherent} provided that
each of its closed, connected subsets is unicoherent.
\enddemo
\demo{ Definition}
A connected topological space $S$ is said to be
{\bf 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.
\enddemo
\demo{ Definition}
A subcontinuum $K$ of a hereditarily unicoherent continuum
$X$ is said to be a {\bf HU-terminal continuum}
of $X$ provided that $K$ is contained in an
irreducible subcontinuum of $X$ and for every
irreducible subcontinuum $I$ of $X$ containing $K$
there is a point $x\in X$ such that
$I$ is irreducible about the union $K\cup \{x\}$.
\enddemo
\demo{ Definition}
Let $X$ be a continuum and $A\subset X$. Then, $X$ is said
to be {\bf irreducible about} $A$ provided no proper
subcontinuum of $X$ contains $A$. A continuum $X$ is said to
be  {\bf irreducible} provided that $X$ is irreducinle about
$\{p,q\}$ for some $p,q \in X$.
\enddemo
\demo{ Definition}
A proper subcontinuum $K$ of a continuum $X$  is said
to be an  {\bf absolutely terminal continuum} of $X$
provided that $K$ is a terminal continuum of
each subcontinuum $L$  of $X$ which properly contains $K$.
\enddemo
\demo{ Definition}
A proper subcontinuum $K$ of a continuum $X$ is
said to be a {\bf terminal continuum} of $X$ provided
that if whenever $A$ and $B$ are proper 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 $X=B\cup K$.
\enddemo
\demo{ Definition}
A continuum that contains no subcontinuum which is itself a
triod is said to be {\bf atriodic}
\enddemo
\demo{ Definition}
A continuum is called a {\bf triod} provided it contains a
subcontinuum whose complement is the union of three nonempty
pairwise disjoint open sets.
\enddemo


\what


\proclaim{\bf Problem 34 }
What mappings among hereditarily unicoherent continua
map HU-terminal subcontinua of the domain onto HU-terminal
subcontinua of the range?
\endproclaim

 (J. J. Charatonik, Absolutely terminal continua
 and confluent mappings, Comment. Math. Univ. Carolinae 32,2
 (1991), p. 382)


\medskip

partially solved
(see
P. Pyrih, T. B\'arta, D. Op\v{e}la, P. R\accent23u\v{z}i\v{c}ka, R. \v{S}\'amal :
Irreducible mappings and HU-terminal continua,
Southwest Journal of Pure and Applied Mathematics 5,2(1999), 31-41.)

\medskip


\demo{ Definition}
A  {\bf continuum} is a nonempty, compact, connected metric
space.
\enddemo
\demo{ Definition}
A connected topological space is said to be
{\bf hereditarily unicoherent} provided that
each of its closed, connected subsets is unicoherent.
\enddemo
\demo{ Definition}
A connected topological space $S$ is said to be
{\bf 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.
\enddemo
\demo{ Definition}
A subcontinuum $K$ of a hereditarily unicoherent continuum
$X$ is said to be a {\bf HU-terminal continuum}
of $X$ provided that $K$ is contained in an
irreducible subcontinuum of $X$ and for every
irreducible subcontinuum $I$ of $X$ containing $K$
there is a point $x\in X$ such that
$I$ is irreducible about the union $K\cup \{x\}$.
\enddemo
\demo{ Definition}
Let $X$ be a continuum and $A\subset X$. Then, $X$ is said
to be {\bf irreducible about} $A$ provided no proper
subcontinuum of $X$ contains $A$. A continuum $X$ is said to
be  {\bf irreducible} provided that $X$ is irreducinle about
$\{p,q\}$ for some $p,q \in X$.
\enddemo





\what


\proclaim{\bf Problem 35 }
Let each nonconstant open mapping on an arcwise connected continuum $X$ is light.
Is the same true for each image of $X$ under a strongly monotone mapping?
\endproclaim

 (P. Pyrih 03/01/1999)

\medskip

SOLVED!
(answer NO, see
P. Pyrih,  D. Op\v{e}la, R. \v{S}\'amal:
 Irreducible mapping and the lightness of open mappings,
 Southwest Journal of Pure and Applied Mathematics,
 to appear)

\medskip



\demo{ Definition}
A mapping $f$ of a space $X$ into a space $Y$ is {\bf open}
if, for each open $A$ in $X$ the image $f(A)$ is open in
$Y$.
\enddemo
\demo{ Definition}
An  {\bf arc} is any space which is homeomorphic to the
closed interval $[0,1]$.
\enddemo
\demo{ Definition}
A  {\bf continuum} is a nonempty, compact, connected metric
space.
\enddemo
\demo{ Definition}
A mapping is {\bf light} if each point inverse is totally
disconnected.
\enddemo
\demo{ Definition}
Let $X$ be a continuum and $A\subset X$. Then, $X$ is said
to be {\bf irreducible about} $A$ provided no proper
subcontinuum of $X$ contains $A$. A continuum $X$ is said to
be  {\bf irreducible} provided that $X$ is irreducinle about
$\{p,q\}$ for some $p,q \in X$.
\enddemo
\demo{ Definition}
Let $X$ be a continuum.
A mapping $f: X \to f(X)=Y$ is said to be
{\bf strongly monotone} provided that for each
nondegenerate irreducible subcontinuum $I\subset Y$ the inverse image $f^{-1}(I)$
is irreducible.
\enddemo


\what


\proclaim{\bf Problem 36 }
All dendrites which are homeomorphic to their open images are openly minimal.
Is the converse true?
\endproclaim

 (J. J. Charatonik, W. J. Charatonik, J. R. Prajs,
 Mapping hierarchy for dendrites,
 Dissertationes Math. (Rozprawy Mat.) (1994), Q2($\Bbb O$), p. 51.)

\medskip

SOLVED!
(answer NO, see
D. Op\v{e}la, P. Pyrih, R. \v{S}\'amal:
 An example of openly minimal dendrite,
 Questions and Answers in General Topology 18,2(2000), 229--236.)

\medskip



\demo{ Definition}
A  {\bf continuum} is a nonempty, compact, connected metric
space.
\enddemo
\demo{ Definition}
A  {\bf simple closed curve} is any space which is
homeomorphic to the unit circle.
\enddemo
\demo{ Definition}
A  {\bf dendrite} means a locally connected continuum containing no
simple closed curve.
\enddemo
\demo{ Definition}
A mapping $f$ of a space $X$ into a space $Y$ is {\bf open}
if, for each open $A$ in $X$ the image $f(A)$ is open in
$Y$.
\enddemo
\demo{ Definition}
A dendrite $X$ is {\bf openly minimal}
provided every open image of $X$ can be openly mapped onto $X$.
\enddemo


\what


\proclaim{\bf Problem 37 }
Does every
chain of dendrites with respect $\le _{\Bbb O}$ have a nondegenerate lower
bound?
\endproclaim

 (J. J. Charatonik, W. J. Charatonik, J. R. Prajs,
 Mapping hierarchy for dendrites,
 Dissertationes Math. (Rozprawy Mat.) (1994), Q4(a)($\Bbb O$), p. 51.)

\medskip

SOLVED!
(answer NO, see
D. Op\v{e}la, P. Pyrih, R. \v{S}\'amal:
 A chain of dendrites openly unbounded from below,
 Mathematica Pannonica 12,1(2001), 83--92.)

\medskip


\demo{ Definition}
A  {\bf continuum} is a nonempty, compact, connected metric
space.
\enddemo
\demo{ Definition}
A  {\bf simple closed curve} is any space which is
homeomorphic to the unit circle.
\enddemo
\demo{ Definition}
A  {\bf dendrite} means a locally connected continuum containing no
simple closed curve.
\enddemo
\demo{ Definition}
A mapping $f$ of a space $X$ into a space $Y$ is {\bf open}
if, for each open $A$ in $X$ the image $f(A)$ is open in
$Y$.
\enddemo
\demo{ Definition}
If $X, Y$ are
dendrites, then $Y \le_O X$ if there is a surjective open mapping $f: X\to Y$.
This quasiordering is not an ordering (i.e. $X \le_O Y \and Y \le_O X$ does not
imply $X$ is homeomorphic to $Y$). To guaratee this we say that $X$ and $Y$
are ${\Bbb O}${\it -equivalent} iff $X \le_O Y$ and $Y\le_O X$ and
consider the quotient of the class of all dendrites. The quasiordering
$\le_O$ induces the ordering $\le_{\Bbb O}$ on the quotient.
\enddemo


\what

\proclaim{\bf Problem 38 }
Does there exist a sequence
$\{X_n : n \in \Bbb N\}$ of dendrites satisfying
$X_{n+1} <_{\Bbb M} X_n$ for every $n\in \Bbb N$?
\endproclaim

 (J. J. Charatonik, W. J. Charatonik, J. R. Prajs,
 Mapping hierarchy for dendrites,
 Dissertationes Math. (Rozprawy Mat.) (1994), Question 5.41(b), p. 17.)

\demo{ Definition}
A  {\bf continuum} is a nonempty, compact, connected metric
space.
\enddemo
\demo{ Definition}
A  {\bf simple closed curve} is any space which is
homeomorphic to the unit circle.
\enddemo
\demo{ Definition}
A  {\bf dendrite} means a locally connected continuum containing no
simple closed curve.
\enddemo
\demo{ Definition}
A continuous function $f:X \to Y$ is said to be  {\bf
monotone} provided  $f^{-1}(y)$ is connected for each $y \in
Y$.
\enddemo

\demo{ Definition}
If $X, Y$ are
dendrites, then $Y \le_M X$ if there is a surjective monotone mapping $f: X\to Y$.
This quasiordering is not an ordering (i.e. $X \le_M Y \and Y \le_M X$ does not
imply $X$ is homeomorphic to $Y$). To guaratee this we say that $X$ and $Y$
are ${\Bbb M}${\it -equivalent} iff $X \le_M Y$ and $Y\le_M X$ and
consider the quotient of the class of all dendrites. The quasiordering
$\le_M$ induces the ordering $\le_{\Bbb M}$ on the quotient.
\enddemo



\what

\proclaim{\bf Problem 39 }
Does there exist a
nondegenerate arcwise  connected continuum such that
each of its proper arcwise connected subcontinua has
empty interior?
\endproclaim

 (W. Makuchowski, J.R. Prajs, 06/01/98)

\demo{ Definition}
A  {\bf continuum} is a nonempty, compact, connected metric
space.
\enddemo


\what

\proclaim{\bf Problem 40 }
Is the locally monotone image of a WAO dendroid a WAO
dendroid?
\endproclaim

 (I. Puga, Mappings on weakly arcwise open dendroids,
Glasnik Matemati\v cki 33(53)(1998), Problem 1, p. 121)

\demo{ Definition}
A continuous function $f:X \to Y$ is said to be  {\bf
monotone} provided  $f^{-1}(y)$ is connected for each $y \in
Y$.
\enddemo
\demo{ Definition}
A continuous function $f:X \to Y$ is said to be  {\bf
locally monotone}
if given any point $p\in X$ there exists a closed
neighborhood $U$ of $p$ such that $f(p)\in$
{Int}$(f(U))$ and $f|U: U \to f(U)$ is a monotone
mapping.
\enddemo
\demo{ Definition}
A  {\bf continuum} is a nonempty, compact, connected metric
space.
\enddemo
\demo{ Definition}
A connected topological space $S$ is said to be
{\bf 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.
\enddemo
\demo{ Definition}
A connected topological space is said to be
{\bf hereditarily unicoherent} provided that
each of its closed, connected subsets is unicoherent.
\enddemo
\demo{ Definition}
A {\bf dendroid} is an arcwise connected, hereditarily unicoherent continuum.
\enddemo
\demo{ Definition}
Given a dendroid $X$ and a subset $W$ of $X$,
we denote by $A(W)$ the set of arc components of $W$.
Let $K$ be a closed subset of $X$.
We say that $X$ is {\bf weakly arcwise open}
(shortly WAO) at $K$ if every $\alpha \in A(X\setminus K)$
is either open or has empty interior;
$X$ is WAO (or $X$ has WAO property)
provided $X$ is WAO at $\{p\}$
for every $p\in X$.
\enddemo



\what


\proclaim{\bf Problem 41 }
Does every
bounded chain of dendrites with respect $\le _{\Bbb O}$ have an infimum (supremum)?
\endproclaim

 (J. J. Charatonik, W. J. Charatonik, J. R. Prajs:
 Mapping hierarchy for dendrites,
 Dissertationes Math. (Rozprawy Mat.) (1994), Q4(a)($\Bbb O$), p. 51.)

\medskip

SOLVED!
(answer NO, see
P. Podbrdsk\'y, P. Pyrih: A chain of dendrites without  open supremum,  Mathematica
Pannonica, to appear;
P. Podbrdský, P. Pyrih: A chain of dendrites without  open infimum,
Southwest Journal of Pure and Applied Mathematics,
submitted.)

\medskip


\demo{ Definition}
A  {\bf continuum} is a nonempty, compact, connected metric
space.
\enddemo
\demo{ Definition}
A  {\bf simple closed curve} is any space which is
homeomorphic to the unit circle.
\enddemo
\demo{ Definition}
A  {\bf dendrite} means a locally connected continuum containing no
simple closed curve.
\enddemo
\demo{ Definition}
A mapping $f$ of a space $X$ into a space $Y$ is {\bf open}
if, for each open $A$ in $X$ the image $f(A)$ is open in
$Y$.
\enddemo
\demo{ Definition}
If $X, Y$ are
dendrites, then $Y \le_O X$ if there is a surjective open mapping $f: X\to Y$.
This quasiordering is not an ordering (i.e. $X \le_O Y \and Y \le_O X$ does not
imply $X$ is homeomorphic to $Y$). To guaratee this we say that $X$ and $Y$
are ${\Bbb O}${\it -equivalent} iff $X \le_O Y$ and $Y\le_O X$ and
consider the quotient of the class of all dendrites. The quasiordering
$\le_O$ induces the ordering $\le_{\Bbb O}$ on the quotient.
\enddemo

\what


\proclaim{\bf Problem 42 }
Does every
bounded chain of dendrites with respect $\le _{\Bbb R}$ have an infimum (supremum)?
\endproclaim

 (J. J. Charatonik, W. J. Charatonik, J. R. Prajs,
 Mapping hierarchy for dendrites,
 Dissertationes Math. (Rozprawy Mat.) (1994), Q4(a)($\Bbb R$), p. 51.)

\medskip

SOLVED!
(answer NO, see
P. Podbrdsk\'y, P. Pyrih: Chain of dendrites without retract infimum,
preprint.)

\medskip


\demo{ Definition}
A  {\bf continuum} is a nonempty, compact, connected metric
space.
\enddemo
\demo{ Definition}
A  {\bf simple closed curve} is any space which is
homeomorphic to the unit circle.
\enddemo
\demo{ Definition}
A  {\bf dendrite} means a locally connected continuum containing no
simple closed curve.
\enddemo
\demo{ Definition}
A~\it mapping \rm means a continuous function. A~surjective
mapping $f: X \to Y\subset X$ is said to be a {\it retraction} provided
$f=f^2$.
\enddemo
\demo{ Definition}
If $X, Y$ are
dendrites, then $Y \le_R X$ if for some $Z\subset X$
homeomorphic to $Y$ there is a retraction $f:
X\to Z$.
This quasiordering is not an ordering (i.e. $X \le_{\Bbb R} Y \and Y \le_{\Bbb R} X$ does not
imply $X$ is homeomorphic to $Y$). To guaratee this we say that $X$ and $Y$
are ${{\Bbb R}}${\it -equivalent} iff $X \le_{\Bbb R} Y$ and $Y\le_{\Bbb R} X$ and
consider the quotient of the class of all dendrites.
\enddemo














\what

\bigskip

\proclaim{\bf Problem 43}
One infinite decreasing sequence of dendrites is known,
find "another" one (i.e. at least one "new" trick ;-)



\endproclaim

 (The known example is in K. Sieklucki : On a family of power $c$ consisting of
$\Re$-uncomparable dendrites,
Fund. Math. 46 (1959), p. 331 -- 335.)

\demo{ Definition}
A  {\bf continuum} is a nonempty, compact, connected metric
space.
\enddemo
\demo{ Definition}
A  {\bf simple closed curve} is any space which is
homeomorphic to the unit circle.
\enddemo
\demo{ Definition}
A  {\bf dendrite} means a locally connected continuum containing no
simple closed curve.
\enddemo
\demo{ Definition}
We say that $X>Y$ provided $X$ contains a topological copy of $Y$ and
$Y$ does not contain a topological copy of $X$.
A sequence  $\{X_n\}_{n=1}^\infty$ is a
{\bf decreasing sequence} provided $X_1 > X_2 > X_3 >
\dots$.
\enddemo


\what

\bigskip

\proclaim{\bf Problem 44}
Find an infinite decreasing sequence of (smooth) fans.
\endproclaim

 (P. Pyrih, 08/08/2001)

\medskip

SOLVED!

\medskip


\demo{ Definition}
The union of arcs emanating from a point v is called a
{\bf fan} provided the singleton {v} is the
intersection of any two of the arcs ; the point v is then
called the {\bf top} of the fan, and the arcs are
called its {\bf arms}.
\enddemo
\demo{ Definition}
A fan $X$ is {\bf smooth} if there is a point $p$ in $X$
such that, if $x_1, x_2, \dots$ is a sequence of points of $X$
converging to the point $x$, then the superior limit of the arcs
$[p,x_n]$ is the arc $[p,x]$.
\enddemo
\demo{ Definition}
We say that $X>Y$ provided $X$ contains a topological copy of $Y$ and
$Y$ does not contain a topological copy of $X$.
A sequence  $\{X_n\}_{n=1}^\infty$ is a
{\bf decreasing sequence} provided $X_1 > X_2 > X_3 >
\dots$.
\enddemo



\what


\proclaim{\bf Problem 45}
When each nonconstant open strongly monotone mapping $f:X \to Y$ defined on a continuum
$X$ gives a continuum $Y$ homeomorphic to $X$?

\endproclaim

 (P. Pyrih 03/01/2001)

\demo{ Definition}
A mapping $f$ of a space $X$ into a space $Y$ is {\bf open}
if, for each open $A$ in $X$ the image $f(A)$ is open in
$Y$.
\enddemo
\demo{ Definition}
A  {\bf continuum} is a nonempty, compact, connected metric
space.
\enddemo
\demo{ Definition}
Let $X$ be a continuum and $A\subset X$. Then, $X$ is said
to be {\bf irreducible about} $A$ provided no proper
subcontinuum of $X$ contains $A$. A continuum $X$ is said to
be  {\bf irreducible} provided that $X$ is irreducinle about
$\{p,q\}$ for some $p,q \in X$.
\enddemo
\demo{ Definition}
Let $X$ be a continuum.
A mapping $f: X \to f(X)=Y$ is said to be
{\bf strongly monotone} provided that for each
nondegenerate irreducible subcontinuum $I\subset Y$ the inverse image $f^{-1}(I)$
is irreducible.
\enddemo


\what

\bigskip

\proclaim{\bf Problem 46}
Does the uniform continuum-chainability of a continuum
imply its arcwise connectedness ?



\endproclaim

 (Iwona Krzemi\' nska :
 Uniform continuum-chainability and Whitney levels, submitted)

\medskip

SOLVED!
(answer NO,
see A. Barkhudaryan, A. Illanes, P. Podbrdsk\'y, P. Pyrih: A uniformly
continuum-chainable continuum which is not arcwise connected, Questions 
and Answers in General Topology, to appear)

\medskip


\demo{ Definition}
A  {\bf continuum} is a nonempty, compact, connected metric
space.
\enddemo
\demo{ Definition}
A  {\bf subcontinuum} is a continuum which is a subset of a
space.
\enddemo
\demo{ Definition}
A continuum $X$ is said to be
{\bf uniformly continuum-chainable} if for each
positive number $\varepsilon$ there is an integer
$k=k(\varepsilon)$ such that for each pair $x$,$y$ of points
of $X$, there are subcontinua $A_1, \dots, A_k$ of $X$ each
of diameter less than $\varepsilon$ such that $x\in A_1$,
$y\in A_k$ and $A_i \cap A_j \ne \emptyset$ whenewer
$|i-j|\leq 1$.
\enddemo
\demo{ Definition}
An  {\bf arc} is any space which is homeomorphic to the
closed interval $[0,1]$.
\enddemo
\demo{ Definition}
A continuum $X$ is said to be
{\bf arcwise connected} if each
pair $x$,$y$ of points
of $X$ is contained in an arc which is contained in $X$.
\enddemo



\what

\bigskip

\proclaim{\bf Problem 47}
Identify the semilattices that are isomorphic to
the semilattice of the compact open sets

(a) of arbitrary spaces

(b) of spaces where they form a sub-base for the topology.



\endproclaim

 (Elias David, Toposym 2001)

\what

\bigskip

\proclaim{\bf Problem 48}
Is there a scattered Valdivia compact space which contains a copy of $[0,\omega_2]$?


\endproclaim

 (O. Kalenda, Toposym 2001)

\demo{ Definition}
A  space is {\bf scattered} if each subset (nonempty)
has an isolated point.
\enddemo
\demo{ Definition}
A  compact space $K$ is {\bf Valdivia} if $K$
is homeomorphic to some $K'\subset [0,1]^\Gamma$
($\Gamma$ is a set, possibly uncountable)
such that
$$
\{x \in K'; \text{supp} X = \{\gamma; x(\gamma)\ne0\}
\text{ is at most countable} \}
$$
is dense in $K'$
\enddemo
\demo{ Definition}
$\omega_2$ is the first ordinal of the second uncountable cardinality.
\enddemo
\demo{ Definition}
$[0,\omega_2]$ is the ordinal segment with the order topology.
\enddemo


\what


\proclaim{\bf Problem 49 }
Has each openly minimal dendrite at most 2 nondegenerate open images?.
Is the converse true?
\endproclaim

 (P. Pyrih 03/03/2001)

\medskip

SOLVED!
(answer NO, see
P. Jel\'\i nek, P. Podbrdsk\'y,
P. Pyrih:
 A~collection of openly minimal dendrites,
 Questions and Answers in General Topology, Vol 19 (2001), 177--182)

\medskip



\demo{ Definition}
A {\bf continuum} is a nonempty, compact, connected metric
space.
\enddemo
\demo{ Definition}
A  {\bf simple closed curve} is any space which is
homeomorphic to the unit circle.
\enddemo
\demo{ Definition}
A  {\bf dendrite} means a locally connected continuum containing no
simple closed curve.
\enddemo
\demo{ Definition}
A mapping $f$ of a space $X$ into a space $Y$ is {\bf open}
if, for each open $A$ in $X$ the image $f(A)$ is open in
$Y$.
\enddemo
\demo{ Definition}
A dendrite $X$ is {\bf openly minimal}
provided every open image of $X$ can be openly mapped onto $X$.
\enddemo

\what

\proclaim{\bf Problem 50}
Find all monotone-homogeneous dendrites.
\endproclaim

  (Open Problem Workshop, Prague 2001)

\demo{ Definition}
A {\bf continuum} is a nonempty, compact, connected metric
space.
\enddemo

\demo{ Definition}
A continuum $X$ is called {\bf monotone-homogeneous} provided that for each
$p,q\in X$ there exist monotone surjective mapping $f:X\to X$ such that
$f(p)=q$.
\enddemo

\demo{ Definition}
A continuous function $f:X \to Y$ is said to be  {\bf
monotone} provided  $f^{-1}(y)$ is connected for each $y \in
Y$.
\enddemo

\demo{ Definition}
A {\bf dendrite} means a locally connected continuum containing no simple
closed curve.
\enddemo

\what

\proclaim{\bf Problem 51}
Does there exist an infinite sequence of dendrites strictly decreasing with
respect to monotone mappings?
\endproclaim

  (Open Problem Workshop, Prague 2001)

\demo{ Definition}
A {\bf continuum} is a nonempty, compact, connected metric
space.
\enddemo

\demo{ Definition}
A {\bf dendrite} means a locally connected continuum containing no simple
closed curve.
\enddemo

\demo{ Definition}
A continuous function $f:X \to Y$ is said to be  {\bf
monotone} provided  $f^{-1}(y)$ is connected for each $y \in
Y$.
\enddemo

\what

\proclaim{\bf Problem 52}
Let $X$ be a continuum, $\dim(X)=1$. Suppose that each monotone image of $X$
has the fixed point property. Must $X$ be a dendrite?
\endproclaim

  (Open Problem Workshop, Prague 2001)

\demo{ Definition}
A {\bf continuum} is a nonempty, compact, connected metric
space.
\enddemo

\demo{ Definition}
A topological space $X$ is said to have the {\bf fixed point property} provided
that for every mapping $f:X\to X$ there exists $x\in X$ such that $f(x)=x$.
\enddemo

\demo{ Definition}
A continuous function $f:X \to Y$ is said to be  {\bf
monotone} provided  $f^{-1}(y)$ is connected for each $y \in
Y$.
\enddemo

\demo{ Definition}
A {\bf dendrite} means a locally connected continuum containing no simple
closed curve.
\enddemo

\what

\proclaim{\bf Problem 53}
Does there exist a model for all planar arcwise connected continua?
\endproclaim

  (Open Problem Workshop, Prague 2001)

\demo{ Definition}
A {\bf continuum} is a nonempty, compact, connected metric
space.
\enddemo

\demo{ Definition}
A continuum $M$ is said to be a {\bf model} for a class $\Cal{C}$ of continua
provided that for every $C\in\Cal{C}$ there exist surjective mapping $f:M\to
C$.
\enddemo

\what

\proclaim{\bf Problem 54}
Find a "good" characterization of being arc-like continuum.
\endproclaim

  (Open Problem Workshop, Prague 2001)

\demo{ Definition}
A {\bf continuum} is a nonempty, compact, connected metric
space.
\enddemo

\demo{ Definition}
If $X$ is a metric space, a mapping $f$ from $X$ to a space
$Y$ is an {\bf $\eps$-map} if, for each point $y$ of
$Y$, $\diam\(f^{-1}\(y\)\) \leq \eps$. If $\Cal C$ is a
collection of continua, a continuum $M$ is {\bf $\Cal
C$-like} if, for every positive number $\eps$, there
exists an $\eps$-map of $M$ onto an element of $\Cal C$.
In particular, a continuum is {\bf tree-like} if, for some
collection $\Cal C$ of trees, $M$ is $\Cal C$-like.
\enddemo

\what

\proclaim{\bf Problem 55}
Let $X$ be a chainable continuum with a mean. Must $X$ be an arc?
\endproclaim

  (Open Problem Workshop, Prague 2001)

\demo{ Definition}
A {\bf continuum} is a nonempty, compact, connected metric
space.
\enddemo

\demo{ Definition}
A {\bf simple chain} is a finite sequence $L_1, L_2, \dots,L_n$
of open sets such that $L_i$ intersects $L_j$ if and only if
$|i-j| \leq 1$.  The terms of the sequence $L_1, L_2, \dots,L_n$
are called the {\bf links} of the chain.
An {\bf $\eps$-chain} is a chain each of whose links has
diameter less than $\eps$.  A continuum $M$ is {\bf
chainable} if, for each positive number $\eps$, $M$ can be
covered by an $\eps$-chain.
\enddemo

\demo{ Definition}
A {\bf mean} on topological space $X$ is a continuous symmetric function
$\mu:X\times X\to X$ which satisfies $\mu(x,x)=x$ for every $x\in X$.
\enddemo

\demo{ Definition}
An  {\bf arc} is any space which is homeomorphic to the
closed interval $[0,1]$.
\enddemo

\what

\proclaim{\bf Problem 56}
Let $X$ be a dendriod with a property of Kelley. Does there exist a mean on
$X$?
\endproclaim

  (Open Problem Workshop, Prague 2001)

\demo{ Definition}
A {\bf continuum} is a nonempty, compact, connected metric
space.
\enddemo

\demo{ Definition}
A {\bf dendroid} is an arcwise connected, hereditarily unicoherent continuum.
\enddemo

\demo{ Definition}
A connected topological space is said to be
{\bf hereditarily unicoherent} provided that
each of its closed, connected subsets is unicoherent.
\enddemo

\demo{ Definition}
A connected topological space $S$ is said to be
{\bf 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.
\enddemo

\demo{ Definition}
A continuum $X$ is said to have {\bf property of Kelley} provided that for
every $a\in X$ and for every subcontinuum $A\subset X$ such that $a\in A$ and
for every sequence $a_n\in X$ which converges to $a$ there exist a sequence of
subcontinua $A_n\subset X$ such that $a_n\in A_n$ and $A_n$ converges to $A$
(in the Hausdorff metric).
\enddemo

\demo{ Definition}
A {\bf mean} on topological space $X$ is a continuous symmetric function
$\mu:X\times X\to X$ which satisfies $\mu(x,x)=x$ for every $x\in X$.
\enddemo

\what

\proclaim{\bf Problem 57}
Does there exist homogeneous continuum with property of Kelley which is
hereditarilly atriodic?
\endproclaim

 (Open Problem Workshop, Prague 2001)

\demo{ Definition}
A {\bf continuum} is a nonempty, compact, connected metric
space.
\enddemo

\demo{ Definition}
A continuum $X$ is said to be {\bf hereditarily atriodic} provided that each
nondegenerate subcontinuum of $X$ is atriodic.
\enddemo

\demo{ Definition}
A continuum that contains no subcontinuum which is itself a
triod is said to be {\bf atriodic}
\enddemo

\demo{ Definition}
A continuum is called a {\bf triod} provided it contains a
subcontinuum whose complement is the union of three nonempty
pairwise disjoint open sets.
\enddemo

\what

\proclaim{\bf Problem 58}
Find a topological characterization of Sierpinski triangle.
\endproclaim

  (Open Problem Workshop, Prague 2001)

\what

\proclaim{\bf Problem 59}
Let $M$ be a continuum such that if $\Cal{G}$ is an uncountable collection of
nondegenerate subcontinua of $M$ then some two elements of $\Cal{G}$ contain a
nondegenerate subcontinuum in their intersection. Does $M$ contain a countable
point set that intersects every nondegenerate subcontinuum of $M$?
\endproclaim

  (Open Problem Workshop, Prague 2001)

\demo{ Definition}
A {\bf continuum} is a nonempty, compact, connected metric
space.
\enddemo

\what

\proclaim{\bf Problem 60}
Let $X$ be a continuum. Let $F_n(X)=\{A\subset X, A\ne\emptyset,|A|\le n\}$,
$F(X)=\bigcup\limits_{n}F_n(X)$. Is $F(X)$ unicoherent for (locally connected)
continuum $X$?
\endproclaim

  (Open Problem Workshop, Prague 2001)

\demo{ Definition}
A {\bf continuum} is a nonempty, compact, connected metric
space.
\enddemo

\demo{ Definition}
A connected topological space $S$ is said to be
{\bf 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.
\enddemo

\what

\proclaim{\bf Problem 61}
Let $X$ be a continuum. Must $X$ be homeomorphic to all its open strongly
monotone images?
\endproclaim

  (Open Problem Workshop, Prague 2001)

\demo{ Definition}
A {\bf continuum} is a nonempty, compact, connected metric
space.
\enddemo

\demo{ Definition}
A mapping $f$ of a space $X$ into a space $Y$ is {\bf open}
if, for each open $A$ in $X$ the image $f(A)$ is open in
$Y$.
\enddemo

\demo{ Definition}
Let $X$ be a continuum and $A\subset X$. Then, $X$ is said
to be {\bf irreducible about} $A$ provided no proper
subcontinuum of $X$ contains $A$. A continuum $X$ is said to
be  {\bf irreducible} provided that $X$ is irreducinle about
$\{p,q\}$ for some $p,q \in X$.
\enddemo

\demo{ Definition}
Let $X$ be a continuum.
A mapping $f: X \to f(X)=Y$ is said to be
{\bf strongly monotone} provided that for each
nondegenerate irreducible subcontinuum $I\subset Y$ the inverse image $f^{-1}(I)$
is irreducible.
\enddemo

\what

\proclaim{\bf Problem 62}
Find all locally connected continua $X$ for which every open monotone
surjective map $f:X\to Y$ is homeomorphism.
\endproclaim

\demo{ Definition}
A {\bf continuum} is a nonempty, compact, connected metric
space.
\enddemo

\demo{ Definition}
A mapping $f$ of a space $X$ into a space $Y$ is {\bf open}
if, for each open $A$ in $X$ the image $f(A)$ is open in
$Y$.
\enddemo

\demo{ Definition}
A continuous function $f:X \to Y$ is said to be  {\bf
monotone} provided  $f^{-1}(y)$ is connected for each $y \in
Y$.
\enddemo

\what

\proclaim{\bf Problem 63}
For continuum $X$ and its subset $A$ denote $T(A)$ the set of all points
$x\in X$ which are not contained in the interior of any subcontinuum $W\subset
X$, $W\cap A=\emptyset$.

Let $X$ be indecomposable continuum, let $A$ be subcontinuum of $X\times X$.
Does it imply $T(W)=X\times X$?
\endproclaim

  (Open Problem Workshop, Prague 2001)

\demo{ Definition}
A {\bf continuum} is a nonempty, compact, connected metric
space.
\enddemo

\demo{ Definition}
A continuum $X$ is said to be  {\bf indecomposable} provided
that $X$ cannot be written as the union of two proper
subcontinua.
\enddemo

\what

\proclaim{\bf Problem 64}
Find all dendrites, which are homeomorphic to all its nondegenerate open
images.
\endproclaim

  (Open Problem Workshop, Prague 2001)

\demo{ Definition}
A {\bf continuum} is a nonempty, compact, connected metric
space.
\enddemo

\demo{ Definition}
A {\bf dendrite} means a locally connected continuum containing no
simple closed curve.
\enddemo

\demo{ Definition}
A mapping $f$ of a space $X$ into a space $Y$ is {\bf open}
if, for each open $A$ in $X$ the image $f(A)$ is open in
$Y$.
\enddemo

\what

\proclaim{\bf Problem 65}
Is the planability of fans preserved by open mappings?
\endproclaim

\demo{ Definition}
A {\bf continuum} is a nonempty, compact, connected metric
space.
\enddemo

\demo{ Definition}
A {\bf fan} is a dendroid with exactly one ramification point.
\enddemo

\demo{ Definition}
A {\bf dendroid} is an arcwise connected, hereditarily unicoherent continuum.
\enddemo

\demo{ Definition}
A connected topological space is said to be
{\bf hereditarily unicoherent} provided that
each of its closed, connected subsets is unicoherent.
\enddemo

\demo{ Definition}
A connected topological space $S$ is said to be
{\bf 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.
\enddemo

\demo{ Definition}
A mapping $f$ of a space $X$ into a space $Y$ is {\bf open}
if, for each open $A$ in $X$ the image $f(A)$ is open in
$Y$.
\enddemo

\enddocument
