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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 the properties of compact connected metric spaces called continua.
It seems that studying continua was for them a major source of new ideas. These
new ideas were later generalized and formed into developed topological
theories. What emerged as continuum
theory is a continuation of this early study of continua. Continuum theory
is not exactly a "theory" separated from other areas of topology and
mathematics by a fixed set of axioms or specific methods. 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 these
problems, continuum theory remains a remarkably fresh area in topology. We
consider these problems 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 these problems.
This is the second edition of our list. The last version of the previous
edition can be found here. Seven problems from the first
edition have been solved. Two of them are from the
classic problems part. Numerous visits in the web site as well as individual
conversations indicate that the list has played its intended role. In the
previous edition, we expressed intention to represent not only our choices but
also, as much as possible, choices of others. After several years of editing of
this list we have realized that this task is impossible to achieve. Individual
involvement in research is so strong that our views on the significance of
particular problems have to be biased. Thus the list we offer is just our
selection of questions we find most interesting and/or important. Nevertheless,
we hope that this web site will continue to serve as source of information for
entire community, specialists as well as non-specialists and students. 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
wjcharat@mst.edu
(573) 341-4909
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. Macías, Topics on continua, Chapman & Hall/CRC,
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
Open problems in topology II, Edited by Elliott Pearl, Elsevier B. V., 2007.
...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 Sierpiński 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 x in X, each subcontinuum K of X containing x and 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.
Yes.
Logan C. Hoehn and Lex G. Oversteegen,
see A complete classification of
homogeneous plane continua, Acta Math. 216 (2016), 177 – 216.
It is known that a positive answer to this question implies such answer to the Question 2.
This question was raised by A. Lelek in Some problems concerning curves, Colloq. Math. 23 (1971), 93-98, Problem 4, p. 94.
4.
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
5.
Is
every nondegenerate, tree-like, homogeneous continuum a pseudo-arc?
6. (SOLVED)
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.
No.
Logan Hoehn, 04-2010, see L. C. Hoehn, A non-chainable planar continuum with span zero, Fund. Math. 211 (2011), 147-174.
7.
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.)
8.
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
9.
(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.
10.
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
Comment:
There was a major attempt to solve this problem by Robert Cauty in 2007.
To the best of our knowledge, his work (see the preprint below) has
neither been published nor confirmed by an independent referee. Therefore the
problem should still be considered open.
Other Problems
No.
Logan
Hoehn, 04-2010, see L. C. Hoehn, An uncountable collection of copies a non-chainable tree-like continuum
in the plane, Proc. Amer. Math. Soc. 141 (2013), 2543-2556.
13.
Is every planar dendroid (arcwise connected continuum) a continuous image of an
arc-like continuum?
This problem is due to J. Krasinkiewicz. It was asked in 1979 and appeared in of Houston Mathematics Problem Book, Problem 155).
The class of continuous images of arc-like continua is a distinctive class known in the literature as the class of weakly chainable continua. They are usually defined by a sequence of “weak chain covers” (see []) and may be characterized as continuous images of the pseudo-arc.
14.
Can any finite dimensional
hereditarily indecomposable continuum be embedded into a finite product of
pseudo-arcs?
(David P. Bellamy)
15.
Is every one-dimensional
pseudo-contractible continuum contractible?
A space X is called pseudo-contractible if there exist a continuum C, two points p0, p1 in C and a map H:X×C→X such that H(x, p0)=x for each x in X and H(X×{ p1}) is constant. The mapping H is called a pseudo-homotopy connecting the identity with a constant map.
(J.R. Prajs, 1995)
20.
Does
there exist 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.
Does
there exist a tree-like continuum that is the image of a continuum under a
2-to-1 map?
(S. B. Nadler, Jr. and L. E.
Ward, 1983)
Remarks
about k-to-1 mappings by Jo Heath
22.
Let
X be a tree-like continuum and let f: X→Y be a map. Is there
an indecomposable subcontinuum W of X such that f(W) intersects W?
(David P. Bellamy)
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,
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.
(SOLVED)
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.
Yes.
Janusz R. Prajs, see Uniformly
path connected homogeneous continua, Topology Proc. 48 (2016), 299–308.
33.
Is
every homogeneous continuum either filament additive or filament connected?
A subcontinuum Y of a space X is called filament provided there is a neighborhood U of Y such that the component of U containing Y has empty interior. A space X is called filament additive if the union of every two filament subcontinua having nonempty intersection is filament. A space X is called filament connected if each two points of X belong to a subcontinuum Y of X such that Y is the union of finitely many filament continua in X .
(J. R. Prajs and K. Whittington, 2005)
34.
Is
every aposyndedic homogeneous curve mutually aposyndedic?
A space X is said to be aposyndetic
provided for every two different points x and y in X
there is a subcontinuum A such
that x is in the interior of A and y in the complement of Y.
If for every two different points x and y in X
there are disjoint subcontinua A
and B containing x and y
in their corresponding
interiors, then X is called mutually aposyndetic.
Aposyndesis was introduced by F. B. Jones in the late 1940s, and mutual aposyndesis by C. L. Hagopian in the late 1960s.
(J. R. Prajs, 2007)