Last Modified
Edited by Janusz R. Prajs
Technical editor Włodzimierz J. Charatonik
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
University of Missouri-Rolla
Rolla, MO 65401
wjcharat@umr.edu
(573) 341-4909
News on the List
October 16, 2006, Problem 33 was positively solved by Alejandro Illanes and Hugo Villanueva, and independently by Mirosław Sobolewski.
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.
May 24, 2004 Problems 20, 21 and remarks by Jo Heath has been added.
March 31, 2004 Problem 9 was solved in the negative by W. J. Charatonik..
March 29, 2004 Problem 27 was added.
April 2, 2003As you can see we have moved to a different account. Please change the address in your favorites list.
March 13, 2003Problem 11 was added.
January 20, 2003Problems 7, 15, and 16, all on homogeneous continua, were added.
August 18, 2002 Problem 33 and an essay by J. J. Charatonik about this problem were added.
June 27, 2002. Problems 13 and 14 appeared on the list. They are related to the study of spaces such as the space of autohomeomorphisms of the Menger continua Mn,k, the space of irrational sequences in the Hilbert space l2, the set of end points in the Lelek fan, the set of end-points in an R-tree. As the research performed by K. Kawamura, L.G. Oversteegen and E.D. Tymchatyn shows, these spaces are topologically very near to each other. The questions were asked by the same, above mentioned authors.
June 22, 2002. Problem 22 was added to the list. This problem was formulated by Lew Ward in connection of the study of continuous selections by E. Michael, and later by S. B. Nadler and F. Jordan.
May 11, 2002. We added Problem 31 to the list. The problem was formulated by Panagiotis Papazoglou, and it appeared in connection with the study of Dr. Papazoglou in geometric group theory. Of course this problem is closely related to Problems 28, 29, and 30 on the list.
In the following books the reader can find basic information about continuum theory:
Illanes and S. B. Nadler, Jr. Hyperspaces, M. Dekker, New York and Basel, 1999.
K. Kuratowski, Topology, vol. 2, Academic Press and PWN, New York, London and Warszawa, 1968.
S. B. Nadler, Jr., Hyperspaces of sets, M. Dekker, New York and Basel, 1978.
S. B. Nadler, Jr., Continuum theory, M. Dekker, New York, Basel and Hong Kong, 1992.
G. T. Whyburn, Analytic topology, Amer. Math. Soc. Colloq. Publ. 28, Providence 1942.
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 Houston problem book. Continua (Cincinnati, OH, 1994), 365–398, Lecture Notes in Pure and Appl. Math., 170, Dekker, New York, 1995. AMS-Tex DVI PDF
W. Lewis, Continuum theory problems, Topology Proc. 8, 1983, 361-394.
Open problems in topology, Edited by Jan van Mill and George M. Reed. North-Holland Publishing Co., Amsterdam, 1990 (H. Cook, W. T. Ingram and A. Lelek, Eleven annotated problems about continua, 295–302; James T. Rogers, Jr., Tree-like curves and three classical problems, 303-310).
…and present
The following concepts are used in the list:
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.
Classical Problems
1.
Does every nonseparating plane (tree-like) continuum have the fixed-point property?
A space X is said to have fixed-point property provided that for every continuous function f: X → X there exists a point p Î X such that f(p)=p.
For more information see the following survey paper: C. L. Hagopian, Fixed-point problems in continuum theory, Continuum theory and dynamical systems (Arcata, CA, 1989), 79–86, Contemp. Math., 117, Amer. Math. Soc., Providence, RI, 1991. MR 92i:54033
2.
(a) Is every nondegenerate, planar, homogeneous, tree-like continuum a pseudo-arc?
(b) Is a confluent image of an arc-like continuum (of a pseudo-arc) necessarily arc-like?
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 S. Mazurkiewicz posed a question as to whether every hereditarily equivalent continuum is an arc [Problème 14, Fund. Math. 2 (1921), 286]. In 1948 E. E. Moise constructed the pseudo-arc, which is hereditarily equivalent and hereditarily indecomposable [An indecomposable plane continuum which is homeomorphic to each of its non-degenerate sub-continua, Trans. Amer. Math. Soc., 63 (1948), 581–594], and thus answered Mazurkiewicz’s question in the negative. Later G. W. Henderson showed that a hereditarily equivalent decomposable continuum is an arc [Proof that every compact decomposable continuum which is topologically equivalent to each of its nondegenerate subcontinua is an arc, Ann. of Math. 72 (1960), 421–428]. H. Cook proved that a hereditarily equivalent continuum is tree-like [Tree-likeness of hereditarily equivalent continua, Fund. Math. 68 (1970), 203–205].
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 Houston problem book. Continua (Cincinnati, OH, 1994), 365–398, Lecture Notes in Pure and Appl. Math., 170, Dekker, New York, 1995, Problem 156, 11/14/79) and, independently, by P. Minc (W. Lewis, Continuum theory problems, Topology Proc. 8 (1983), 361–394, Problem 81, p. 379
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 ?
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)
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
11.
Suppose M1, M2,… is a sequence of mutually disjoint continua in the plane converging to the continuum M homeomorphically. Is M circle-like or chainable?
The statement that the sequence M1, M2,… converges homeomorphically to the continuum M means there exists a sequence h1, h2,… of homeomorphisms such that, for each positive integer i, hi is a homeomorphism from Mi onto M and for each positive number ε there exists a positive integer N such that if j > N then, for all x, dist(hj(x),x) < ε.
The problem was stated by J. B. Fugate in 1978 in University of Houston Mathematics Problem Book, Problem 107.
12.
Let X be a nondegenerate continuum such that the plane admits a continuous decomposition into topological copies of X. Must then X be hereditarily indecomposable? Must X be the pseudo-arc?
The existence of a continuous decomposition of the plane into pseudo-arcs was announced by R. D. Anderson in 1950. The first known proof of this fact appeared in [W. Lewis and J. J. Walsh, A continuous decomposition of the plane into pseudo-arcs, Houston J. Math. 4 (1978), 209-222].
The second part of this problem was formulated by J. Krasinkiewicz in 1979 (see University of Houston Mathematics Problem Book, Problem 158).
13 (SOLVED).
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. Steprãns, Complete Erdös space is unstable, Math. Proc. Cambridge Philos. Soc. 137 (2004), 465-473, Corollary 4.2, p. 469.
14 (SOLVED).
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. Steprãns, Complete Erdös space is unstable, Math. Proc. Cambridge Philos. Soc. 137 (2004), 465-473, Corollary 3.2, p. 467.
15.
Let X be a nondegenerate homogeneous continuum. Must X topologically contain either an arc, or a nondegenerate, hereditarily indecomposable continuum?
This problem is related to Problems 7 and 16. (J. R. Prajs, December 2002)
16.
Let X be a nondegenerate homogeneous continuum such that every hereditarily indecomposable subcontinuum of X is degenerate. Is X a solenoid?
This problem is related to Problems 7 and 15. (J. R. Prajs, December 2002)
17 (SOLVED).
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
18.
Suppose there is a continuous surjection f: X → Y between continua X and Y. Does there then exist a continuous surjection between the corresponding hyperspaces C(X) and C(Y) of subcontinua?
(J.R. Prajs, 1995)
19.
Suppose there is a continuous surjection f: X2 → Y2 between Cartesian squares of continua X and Y, correspondingly. Does there then exist a continuous surjection from X onto Y ?
(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-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.
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, March 21, 2002)
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, March 21, 2002)
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, May 11, 2002)
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)
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.