1^{st}
Edition Solved Problems

Last Modified Thursday, 09-Mar-2017 18:42:30 CST

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)*