Open
Problems in Continuum Theory, 2nd Edition
Last Modified Monday, 20-Dec-2010 15:23:47 CST
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 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
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 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.
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 in 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
An essay about this problem by Charles L. Hagopian
2.
Is every nondegenerate, planar, homogeneous, tree-like continuum a
pseudo-arc?
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.
3. Is a confluent image of an arc-like continuum (of a pseudo-arc)
necessarily arc-like?
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.
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.
Other Problems
10. (SOLVED).
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
No.
Logan Hoehn, 04-2010. If X is the continuum described in L. Hoehn, A nonchainable plane continuum with span zero, Preprint pdf, then X×Cantor set is embeddable in the plane.
11.
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
12.
Is every planar dendroids
(arcwise connected continuum) a continuous image of an arc-like continumm?
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.
13.
Can any finite dimensional
hereditarily indecomposable continuum be embedded into a finite product of
pseudo-arcs?
(David P. Bellamy)
14.
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.
The problem was stated by W. Kuperberg in
1972 in University of Houston Mathematics Problem Book, Problem 29.
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 8 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 8 and 15.
(J. R. Prajs, December 2002)
17.
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)
18.
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)
19.
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)
20.
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
21.
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)
22.
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)
23.
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)
24.
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)
25.
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)
26.
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.
27.
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)
28.
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)
29.
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,
30.
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,
31.
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.
32.
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)
33.
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)
