# Open problems in continuum theory

These problems are the second edition of problems posted at this webpage maintained by Włodzimierz J. Charatonik and Janusz R. Prajs.

Problem 1 (OPICT): Does every nonseparating plane (tree-like) continuum have the fixed-point property?

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, 1991.

Problem 3 (OPICT): Is a confluent image of an arc-like continuum (of a pseudo-arc) necessarily arc-like?

Problem 4 (OPICT): 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?

Problem 6 (OPICT): Let $X$ be a continuum with span zero. Must $X$ be arc-like?

Answer: No. Logan Hoehn, 04-2010, see L. C. Hoehn, A non-chainable planar continuum with span zero, Fund. Math. 211 (2011), 147-174. 

Problem 9 (OPICT): Let $X$ be a homogeneous, $n$-dimensional continuum. If $X$ is an absolute neighborhood retract, must $X$ be an $n$-manifold?

Answer: It has been proven by Bing and Borsuk that it is true for $n<3$.

Problem 10 (OPICT): Suppose $M_1,M_2,\ldots$ is a sequence of mutually disjoint continua in the plane converging to the continuum $M$ homeomorphically. Is $M$ circle-like or chainable?

Answer: 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.

Problem 11 (OPICT): 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?

Answer: 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].

Problem 12 (OPICT): Is every planar dendroid a continuous image of an arc-like continuum?

Problem 13 (OPICT): Can any finite dimensional indecomposable continuum be embedded into a finite product of pseudo-arcs?

Problem 15 (OPICT): Let $X$ be a nondegenerate homogeneous continuum. Must $X$ topologically contain either an arc, or a nondegenerate, hereditarily indecomposable continuum?

Problem 16 (OPICT): Let $X$ be a nondegenerate homogeneous continuum such that ev ery hereditarily indecomposable subcontinuum of $X$ is degenerate. Is $X$ a solenoid?

Problem 17 (OPICT): Suppose there is a continuous surjection $f \colon X \rightarrow Y$ between continua $X$ and $Y$. Does there then exist a continuous surjection between the corresponding hyperspaces $C(X)$ and $C(Y)$ of subcontinua?

Problem 18 (OPICT): Suppose there is a continuous surjection $f \colon X^2 \rightarrow Y^2$ between Cartesian squares of continua $X$ and $Y$, correspondingly. Does there then exist a continuous surjection from $X$ onto $Y$?

Problem 19 (OPICT): Does there exist a two-to-one map defined on the pseudo-arc?

Problem 20 (OPICT): Does there exist a tree-like continuum that is the image of a continuum under a two-to-one map?

Problem 21 (OPICT): Let $X$ be a tree-like continuum and let $f \colon X \rightarrow Y$ be a map. Is there an indecomposable subcontinuum $W$ of $X$ such that $f(W)$ intersects $W$?

Problem 22 (OPICT): Let $X$ be an absolute retract for hereditarily unicoherent continua. Must $X$ be a tree-like continuum? Must $X$ have the fixed point property?

Problem 24 (OPICT): Let $X$ be an atriodic absolute retract for hereditarily unicoherent continua. Must $X$ be the inverse limit of arcs with open bonding mappings?

Answer: Such a continuum $X$ must be an indecomposable, arc-like, Kelley continuum with only arcs for proper subcontinua.

J. J. Charatonik, W. J. Charatonik and J. R. Prajs, Atriodic absolute retracts for hereditarily unicoherent continua, Houston J. Math. 30 (2004), 1069 - 1087

J. J. Charatonik and J. R. Prajs, Generalized ε-push property for certain atriodic continua, Houston J. Math. 31 (2005), 441-450.

Problem 25 (OPICT): Let $B_3$ be the $3$-book. Does $B_3$ admit a continuous decomposition into pseudo-arcs?

Problem 26 (OPICT): Let $T$ be a simple triod. Do there exist maps $f,g \colon T \rightarrow T$ such that $fg=gf$ and $f(x) \neq g(x)$ for each $x \in T$?

Problem 27 (OPICT): Does there exist a nondegenerate, homogeneous, locally connected continuum $X$ in the $3$-space $\mathbb{R}^3$ that is topologically different from a [[circle], the Menger curve, a $2$-manifold and from the Pontryagin sphere?

Problem 28 (OPICT): Let $X$ be a simply connected, nondegenerate, homogeneous continuum in the $3$-space $\mathbb{R}^3$. Must $X$ be a homeomorphic to the unit sphere?

Problem 29 (OPICT): Let $X$ be a simply connected, homogeneous continuum. Must $X$ be locally connected?

Problem 30 (OPICT): Let $X$ be a homogeneous, simply connected (locally connected) nondegenerate continuum. Must $X$ contain a $2$-dimensional disk?
Problem 31 (OPICT): Let $X$ be an arcwise connected, homogeneous continuum. Must $X$ be uniformly path connected? (Equivalently, is $X$ a continuous image of the Cantor fan?)