Solved
Problems
Last Modified
Edited by
Janusz R. Prajs
Technical editor Włodzimierz J. Charatonik
Janusz R. Prajs
Department of Mathematics and Statistics
California State University, Sacramento
6000 J Street
Sacramento, CA 95819-6051
prajs@csus.edu
(916) 278-7118
or
Włodzimierz
J. Charatonik
Department of Mathematics and Statistics
Missouri University
of Science and Technology
Rolla, MO
65401
wjcharat@mst.edu
(573) 341-4909
Here
we present problems from the past edition of our list that have been solved.
However, we cannot guarantee that all the announced results are correct; the
authors are ultimately responsible for the correctness of their solutions.
1 .
If X is a Kelley continuum, is the hyperspace C(X)
of nonempty subcontinua of X also a Kelley continuum?
(S. B. Nadler, Jr,
1978)
An essay about this problem by Włodzimierz J. Charatonik
NO
Wlodzimierz
J. Charatonik with Janusz J. Charatonik, March 2004.
Janusz J. Charatonik and Włodzimierz J. Charatonik,
Property of Kelley for
the Cartesian products and hyperspaces, Proc. Amer. Math. Soc.
136 (2008), no. 1,
341-346.
2.
Let H be the space of all autohomeomorphisms of the Menger
curve M1,3 (of the Menger space Mn,k, for 0 < n < k). Is H
homeomorphic to the complete Erdös space E ?
We assume the regular sup metric in
the spaceH. The complete Erdös
space E is defined as the collection of all sequences of irrational
numbers xn such that the series Σxn2
converges, equipped with the metric d({xn},{yn}) = (Σ(xn-yn)2)1/2.
Observe that the complete Erdös
space is the subspace of sequences of irrational numbers in the Hilbert space l2.
Oversteegen and Tymchatyn
conjectured that the answer is YES (Conjecture
7.8 in J. C. Mayer and L. G. Oversteegen., Continuum
theory, Recent Progress in General
Topology North-Holland, Amsterdam, 1992), pp. 453--492.). This question is
closely related to the next problem on the list.
NO
Jan J. Dijkstra,
J. van Mill, and J. Steprăns, Complete Erdös
space is unstable, Math. Proc. Cambridge
Philos. Soc. 137 (2004), 465-473, Corollary 4.2, p. 469.
3 .
Let X be an almost 0-dimensional,
1-dimensional, topologically complete, pulverized, homogeneous space. Is
X homeomorphic to the complete Erdös space?
Let uoc(X)
be the collection of all unions of open-closed subsets of X. If the
collection of all open sets U in X such that X - cl(U)
belongs to uoc(X) is a basis of the
topology on X, then we say that X is almost 0-dimensional.
We call X a pulverized space if it is homeomorphic to a space Y
- {p}, where p is a point in a connected metric space Y.
This problem is related to the previous one.
The question was originally asked by Kazuhiro Kawamura, Lex G. Oversteegen and Edward D. Tymchatyn
in On homogeneous totally disconnected 1-dimensional spaces, Fund. Math.
150 (1996), 97-112. Among spaces that have the same properties as the space X
in the problem are: the complete Erdös space, the
space of autohomeomorphisms H of any Menger continuum Mn,k
(k>0), the set of end points of the Lelek
fan, the set of end points of the universal R-tree, the set of end points of
the Julia set of the exponential map (see the paper by Kawamura, Oversteegen and Tymchatyn
mentioned above, and also: L.G. Oversteegen and E.D. Tymchatyn, On the dimension of certain totally
disconnected spaces, Proc. Amer. Math. Soc. 122 (1994), 885--891). Among
totally disconnected 1-dimensional spaces, the complete Erdös
space seems to play a role similar to the role of irrationals among 0-dimensional
spaces.
NO
Jan J. Dijkstra,
J. van Mill, and J. Steprăns, Complete Erdös
space is unstable, Math. Proc. Cambridge
Philos. Soc. 137 (2004), 465-473, Corollary 3.2, p. 467.
4 .
Does there exist a continuum X that
does not admit a continuous surjection onto its hyperspace C(X) of
nonempty subcontinua?
Originally, questions about the existence of
such mappings appeared in the book by S. B. Nadler, Jr., Hyperspaces of
sets, M. Dekker, New York and Basel, 1978.
YES
Alejandro Illanes, August 31, 2004
Alejandro Illanes , A continuum whose hyperspace of
subcontinua is not its continuous image, Proc. Amer.
Math. Soc. 135 (2007),
no. 12, 4019-4022
5.
Let
X be a dendrite and C be a collection of closed, nonempty,
mutually disjoint subsets of X. Assuming the Hausdorff
metric on C, does C admit a continuous
selection s: C → X ?
This question was posed by Lew
Ward during the 8-th Chico Topological Conference, May 30-June 1, 2002. Originally, the property
in question was studied by Ernest Michael (Topologies on spaces of subsets,
Trans. Amer. Math. Soc. 71, (1951). 152--182) as what he called property S4.
He observed that an S4 space cannot contain a simple closed curve and that any
tree is S4. There was no further progress on this problem until 1999 when Sam
B. Nadler and Francis Jordan proved (A result about a selection problem of
Michael, Proc. Amer. Math. Soc. 129 (2001), 1219--1228) that an S4
continuum is hereditarily decomposable. Finally, Jordan proved (preprint) that an S4
continuum is a dendrite. Thus in the above question L. Ward asks whether the
converse implication to Jordan's
result is true.
YES
Francis Jordan, July 22, 2004.
Francis Jordan, The S4 continua in the sense of Michael are precisely the
dendrites, Houston J. Math. 32 (2006), no. 2,
471—487.
6 .
Is
there an uncountable family of dendroids each two
members of which are incomparable by continuous functions?
A dendroid is an arcwise connected and hereditarily unicoherent metric
continuum.
(B. Knaster 1961)
YES
Announced by Piotr
Minc, February, 2006.
Preprint
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