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In the first half of the twentieth century,
when foundations of general topology had been established, many famous
topologists were particularly interested in properties of compact connected
metric spaces, called continua. What later emerged as continuum
theory is a continuation of that study. Continuum theory not exactly is a
"theory" separated from other areas of topology and mathematics, and
its identity is rather defined by special type of questions asked in this area.
Now, when basic general topology is already established, many deep but
naturally and simply formulated problems in continuum theory still remain open.
Due to those problems continuum theory remains remarkably fresh among other
areas of topology. We consider those problems very interesting and important.
We think that it would be useful to find a place where they could be
continuously exposed and updated. Therefore we have decided to present this web
site so that everyone interested, especially beginners, can find them together
with some basic information necessary to start working on those problems. We
present the problems in the selection and ordering (starting with the most
important ones) according to editor's opinion. We understand that, perhaps,
other specialists would choose different selections. Nevertheless, the list
below is intended to also include preferences of others. The reader will judge
how far we are successful in this effort. By its very nature the work of
preparing such a site is never complete and should be updated continuously. We
welcome all comments and suggestions from the reader to help in preparing this
web page. If you have some important information about any particular problem
or you believe that some problem should be added to the list, please contact
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
wjcharat@umr.edu
(573) 341-4909
February 11,
2006, Problem 34 was positively solved
by Piotr Minc.
November 11,
2005, Problems 24 and 25 were added.
August 31, 2004, Problem 34 was added.
July 22, 2004, Problem 22 was positively solved by Francis Jordan.
March 31, 2004 Problem 9 was solved in the negative by W. J. Charatonik..
March 29, 2004 Problem 27 was added.
August 18, 2002 Problem 33 and an essay by J. J. Charatonik about this problem were added.
In the following books the reader can find basic information about continuum theory:
Illanes and S.
B. Nadler, Jr. Hyperspaces, M. Dekker,
K. Kuratowski, Topology,
vol. 2, Academic Press and PWN,
S. B. Nadler,
Jr., Hyperspaces of sets, M. Dekker,
S. B. Nadler,
Jr., Continuum theory, M. Dekker,
G. T. Whyburn, Analytic
topology, Amer. Math. Soc. Colloq. Publ. 28,
A lot of information about continuum theory, and many definitions can be found on the web page Examples in continuum Theory by Janusz. J. Charatonik, Pawel Krupski and Pavel Pyrih.
We also give references to other lists of continuum theory problems published in the past:
H. Cook, W. T.
Ingram, A. Lelek, A list of problems known as
W. Lewis, Continuum theory problems, Topology Proc. 8, 1983, 361-394.
Open problems
in topology, Edited by
...and present
A compact, connected Hausdorff space is called Hausdorff continuum. By a continuum we mean a compact, connected metric space.
If ε > 0 is a positive number and f: X → Y is a continuous function between metric spaces X and Y and diam f -1(y)< ε > 0 for each y in Y, then f is called an ε -map. A connected, acyclic graph is called a tree. A continuum admitting, for every ε > 0 an ε-map onto a tree (onto the unit segment [0,1]) is said to be tree-like (arc-like).
A continuum X is called unicoherent provided that for every pair A, B of subcontinua of X such that X is the union of A and B, the intersection of A and B is connected. If every subcontinuum of a continuum X is unicoherent, then X is called hereditarily unicoherent. All tree-like continua are hereditarily unicoherent. A hereditarily unicoherent, arcwise connected continuum is called a dendroid. All dendroids are known to be tree-like. A locally connected dendroid is called a dendrite. Equivalently, a locally connected continuum X is a denrite if and only if X contains no simple closed curve. Another equivalent condition is that X is a compact absolute retract for metrizable spaces and dim X < 2.
A space X is called homogeneous if and only if for every pair of points x, y Î X there exists a homeomorphism h : X → X such that h(x)=y.
For any metric space X the symbol C(X) denotes the collection of all nonempty subcontinua of X equipped with the Hausdorff metric.
Let k, n be positive integers with k < n and Mn,k be the n-dimensional Menger continuum in the Euclidean space Rk such that Mn,k is universal among all n-dimensional compacta embeddable into Rk (K. Menger, Kurventheorie, Teubner, Leibzig, 1932). The construction of spaces Mn,k can be sketched as follows. Let X1 be the cube [0,1]k naturally embedded in Rk. We represent X1 as the union of 3k congruent smaller cubes according to the decomposition of [0,1] into the intervals [0,1/3], [1/3,2/3], [2/3,1]. Among the smaller cubes we select those which intersect the n-dimensional skeleton of[0,1]k. Let X2 be the union of all selected smaller cubes. For each selected smaller cube K let K′ be the subset of K such that the pairs (K′,K) and (X2,X1) are geometrically similar. Let X3 be the union of all such sets K' for all selected smaller cubes K. In the similar manner we define a nested sequence of compacta Xm for m=1,2,... . The Menger space Mn,k is defined as the intersection of the sequence of sets X1, X2, ... . Note that M1,2 is the Sierpinski universal plane curve, M1,3 is the Menger universal curve, and, if we also admit n=0, the space M0,1 is the Cantor set.
A continuum X is called a Kelley continuum provided that for each point for each subcontinuum K of X containing x and for each sequence of points xn converging to x there exists a sequence of subcontinua Kn of X containing xn and converging to the continuum K.
The study of homogeneous continua was initiated by the question whether every planar, homogeneous, nondegenerate continuum is homeomorphic to a circle, posed by K. Kuratowski and B. Knaster in Problème 2, Fund. Math. (1920), 223. For the definition of the pseudo-arc and for more information about this continuum see W. Lewis, The pseudo-arc, Bol. Soc. Mat. Mexicana (3), vol. 5 (1999), 25-77.
It is known that a positive answer to the question (b) implies such answer to the question (a).
The question (b) was raised by A. Lelek in Some problems concerning curves, Colloq. Math. 23 (1971), 93-98, Problem 4, p. 94.
3.
Assume that a nondegenerate
continuum X is homeomorphic to each of its nondegenerate subcontinua.
Must then X be either an arc or a pseudo-arc?
Continua
homeomorphic to every of their nondegenerate subcontinua are named hereditarily
equivalent. As early as 1921
4.
Is every nondegenerate, tree-like, homogeneous
continuum a pseudo-arc?
5.
Let X be a continuum
with span 0. Must X be arc-like?
For any two maps
f,g: Z → Y, where Y is a metric space, define m(f,g)= inf{d(f(z),g(z))|
z Î Z}. For any
continuum X the number σ(X)= sup{ m(f,g)|f,g: Z →
X, where Z is a continuum, and f(Z) Í g(Z) } is called the span of X. Note
that σ(X)=0 is a topological property of a continuum X. The
concept of the span of a continuum is due to Andrzej Lelek.
The above question was posed by A. Lelek in Some problems concerning curves, Colloq. Math. 23 (1971), 93-98.
6.
Does every nondegenerate,
homogeneous, indecomposable continuum have dimension 1?
This questions was asked by James. T. Rogers, Jr. In the nonmetric case the answer is negative (J. van Mill, An infinite-dimensional homogeneous indecomposable continuum, Houston J. Math. 16 (1990), 195--201.)
7.
Is every hereditarily decomposable, homogeneous
nondegenerate continuum a simple closed curve?
This questions was
asked by J. Krasinkiewicz, (H. Cook, W. T. Ingram, A. Lelek A list of
problems known as
8. (SOLVED).
Is it true that for each
dendroid X and for each ε > 0 there is a tree Tε
contained in X and a retraction rε: X→ Tε
with d(x,rε(x)) < ε for each x Î X ?
An essay about this problem by Janusz J. Charatonik in pdf format
YES
Announced by Robert Cauty.
9 (SOLVED).
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 in pdf format
NO
Wlodzimierz
J. Charatonik with Janusz J. Charatonik, 03-31-2004.
10.
(R.H. Bing, K. Borsuk) Let X be a
homogeneous, n-dimensional continuum. If X is an absolute
neighborhood retract (ANR), must X be an n-manifold?
A positive answer to this question was given by Bing and Borsuk for n < 3.
Other Problems
NO
Jan J. Dijkstra, J. van Mill, and J. Steprãns, Complete Erdös space is unstable, Math.
Proc.
NO
Jan J. Dijkstra, J. van Mill, and J. Steprãns, Complete Erdös space is unstable, Math.
Proc.
YES
(J.R. Prajs, 1995)
20.
Is
there a 2-to-1 map defined on the pseudoarc?
A map is called 2-to-1 if preimage of every point has exactly two points.
(J. Mioduszewski 1961)
21.
Is there a tree-like continuum that is the image of a 2-to-1 map from a continuum?
(S. B. Nadler, Jr. and L. E.
Ward, 1983)
Remarks about k-to-1 mappings by Jo Heath
22
(SOLVED).
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-
YES
Francis Jordan, July 22, 2004.
23.
Let
X be an absolute retract for hereditarily unicoherent continua. Must X
be a tree-like continuum? Must X have
the fixed point property?
(J.J. Charatonik, W.J. Charatonik, J.R. Prajs,
1998)
24.
Is
each Kelley dendroid an absolute retract for hereditarily unicoherent continua?
If such a dendroid is an inverse limit of trees with conflunet bonding maps, then it is an absolute retract for hereditarily unicoherent continua (see J. J. Charatonik, W. J. Charatonik and J. R. Prajs, Hereditarily unicoherent continua and their absolute retracts, Rocky Mountain J. Math. 34 (2004), 83 - 110).
(J.J. Charatonik, W.J. Charatonik, J.R. Prajs,
1998)
25.
Let
X be an atriodic absolute retract for hereditarily unicoherent continua.
Must X be the inverse limit of arcs with open bonding mappings?
Such a continuum X must be an indecomposable, arc-like, Kelley continuum with only arcs for proper subcontinua. These results can be found in the following two articles: J. J. Charatonik, W. J. Charatonik and J. R. Prajs, Atriodic absolute retracts for hereditarily unicoherent continua, Houston J. Math. 30 (2004), 1069 - 1087, and, J. J. Charatonik and J. R. Prajs, Generalized ε-push property for certain atriodic continua, Houston J. Math. 31 (2005), 441--450.
(J.J. Charatonik, W.J. Charatonik, J.R. Prajs,
1998)
26.
Let
B3 be the 3-book, i.e. the product of the closed interval
[0,1] and a simple triod T. Does B3 admit a continuous
decomposition into pseudo-arcs?
All locally connected continua without local separating points that are embeddable in a surface admit a continuous decomposition into pseudo-arcs [J. R. Prajs, Continuous decompositions of Peano plane continua into pseudo-arcs, Fund. Math. 158 (1998), 23-40] and the Menger universal curve also admits such a decomposition [J. R. Prajs, Continuous decompositions of the Menger curve into pseudo-arcs, Proc. Amer. Math. Soc. 128 (2000), 2487-2491]. The only known obstacle that prevents a construction of such a decomposition of a locally connected continuum is a local separating point. However the methods developed in the two above papers cannot be generalized to all locally connected continua without local separating point. The 3-book seems to be one of the simplest examples of such continua, for which those methods failed.
(J.R. Prajs, 1997)
27.
Let
T be a simple triod. Do there exist maps f,g:T →T such that
fg=gf and f(x)≠g(x) for each x in T ?
Positive answer to this question would allow a construction of a
(simple triod)-like continuum admitting a fixed point free map. No such example
is known so far. Negative answer wold generalize the fixed point property of
the simple triod. It is interesting whether such maps exist for trees other
than a simple triod. This question was asked in 1970’s or 1980’s. The original
author of the question is unknown.
28.
Does
there exist a nondegenerate, homogeneous, locally connected continuum X
in the 3-space R3 that is topologically different from a circle, the
Menger curve, a 2-manifold and from the Pontryagin sphere?
It is known that such a continuum X must have dimension 2, cannot be an ANR and it cannot topologically contain a 2-dimensional disk.
(J.R. Prajs, 1996)
29.
Let
X be a simply connected, nondegenerate, homogeneous continuum in the
3-space R3. Must X be homeomorphic to the unit sphere S2
?
A continuum X is called simply connected provided that X is arcwise connected and every map from the unit circle S1 into X is nulhomotopic. If X either is an ANR, or topologically contains a 2-dimensional disk, then the answer is YES.
(J. R. Prajs,
30.
Let
X be a simply connected, homogeneous continuum. Must X be locally
connected?
This question is related to a question by K. Kuperberg whether an arcwise connected, homogeneous continuum must be locally connected.This last question was recently answered in the negative by J. Prajs.
(J.R. Prajs,
31.
Let
X be a homogeneous, simply connected (locally connected) nondegenerate
continuum. Must X contain a 2-dimensional disk?
This question appeared in connection with the study of Panagiotis Papazoglou in geometric group theory.
(P. Papazoglou,
32.
Let
X be an arcwise connected, homogeneous continuum. Must X be
uniformly path connected? (Equivalently, is X a continuous image of the
Cantor fan?)
A continuum X is called uniformly
path connected provided that there is a compact collection P of
paths in X such that each pair of points x, y in X is
connected by some member of P. The Cantor fan is defined as the
cone over the Cantor set. It is known that a homogeneous arcwise connected
continuum need not be locally connected (J. R. Prajs, A homogeneous arcwise
connected curve non-locally-connected curve, American J. Math. 124 (2002),
649-675). The strongest result in the direction of this question has been
obtained by D. P. Bellamy, Short paths in homogeneous continua, Topology
Appl. 26 (1987), 287-291. See also: D.P. Bellamy, Arcwise connected
homogeneous metric continua are colocally arcwise connected, Houston J.
Math. 11 (1985), 277-281, and D.P. Bellamy and L. Lum, The cyclic
connectivity of homogeneous arcwise connected continua, Trans. Amer. Math.
Soc. 266 (1981), 389-396.
33
(SOLVED).
Is
the arc the only arc-like continuum that admits a mean?
A mean on a space X is defined as a continuous mapping μ: X × X → X such that μ(x,y)=μ(y,x) and μ(x,x) = x for every x,y Î X.
(P. Bacon, 1969)
An essay about this problem by
Janusz J. Charatonik in pdf format
YES
Alejandro Illanes and Hogo Villanueva, September 2006
And independently
Mirosław Sobolewski, September 2006
34
(SOLVED).
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
Piotr Minc, February, 2006.