1^{st}
Edition Solved Problems

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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.

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*.

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.

12.

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.

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

*(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

*(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
*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) *

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 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)*

28.

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) *

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, March 21, 2002) *

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,
May 11, 2002) *

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)*