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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 * 365-398, Lecture Notes in Pure and Appl. Math., 170,

W. Lewis, *Continuum theory
problems,* Topology Proc. 8, 1983, 361-394.

*Open problems in topology,*
Edited by *Eleven annotated problems about
continua,* 295-302; James T. Rogers, Jr., *Tree-like curves and three
classical problems,* 303-310).

*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 *M _{n,k}* be
the

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* x _{n}*
converging to

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 *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].

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 * 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,

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 *T _{ε}*
contained in

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 *p _{0}, p_{1} *in

*(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

*(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
*B _{3}* be the 3-book, i.e. the product of the closed interval
[0,1] and a simple triod

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 R^{3} 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 R^{3}. Must *X* be homeomorphic to the unit sphere *S ^{2}*
?

A continuum *X* is called *simply
connected* provided that *X* is arcwise
connected and every map from the unit circle *S ^{1}* into

*(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.
(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)*