36 problems

A treelike continuum is a continuum whose every subcontinuum is unicoherent and whose nondegenerate subcontinua are arc-like. A map $f$ is strictly locally 1-to-1 if it is locally…

Dimension-one conjecture. Every co-existentially closed continuum has Lebesgue covering dimension one.

Let $N$ be a non-planar tree-like continuum, and let $\widetilde N$ denote its deleted product with the involution exchanging the two factors. Tree-like continuum conjecture. There…

Non-density conjecture. The periodic homeomorphisms, and likewise the almost periodic or pointwise-periodic homeomorphisms, are not dense in $\operatorname{Homeo}(P)$.

Watson's conjecture. Such an example exists in ${\sf ZFC}$, probably even a completely regular one, and its existence depends on hard finite combinatorics.

Let $P$ be the pseudoarc, the unique hereditarily indecomposable chainable continuum, and let $G=H(P)$ be its homeomorphism group. Let $M_G$ denote the universal minimal compact…

Let $X$ be a non-degenerate compact metric space, and suppose that every midset of $X$ is homeomorphic to the $(n-1)$-sphere $S^{n-1}$. Higher-dimensional double midset conjecture.…

A continuum is a non-degenerate connected compact metric space. For a metric space, the double midset property (DMP) means that every pair of distinct points has a midset consistin…

Finitely generatedness conjecture. Any continuously homogeneous indecomposable planar continuum is finitely generated.

Conjecture on finitely generated continua. There is a large and quite natural class of continua that lack continuous homogeneity.

Charatonik's conjecture. The pseudo-circle might provide another counterexample to that equivalence.

Core decomposition equality. The core decomposition of $K$ satisfies

Euler characteristic conjecture. We conjecture that

Classification conjecture. If $EPG(F)=EPG(G)\neq\0$, then $F$ and $G$ are homeomorphic.

Let $f:M^1_3\to\mathbb R^3$ be the embedding from Proposition 3-tangled, let $L\subset\mathbb R^3$ be the corresponding closed broken line, and set … Property 4 means that it is im…

Let $\mathbb{G}\mathbb{S}_2$ denote the Griffiths double cone space and let $\mathbb{H}\mathbb{A}$ denote the harmonic archipelago. For a pointed space $X$, write $\pi_1(X)$ for it…

The minimal decomposition conjecture. The point $p$ is non-coastal when treated as a point of each $X_n$. This asserts the desired non-coastal behavior for every minimal decomposit…

Let $X$ be a continuum with no shore points, let $p\in X$, and let $X=X_1\oplus\cdots\oplus X_N$ denote a decomposition into subcontinua. A point is non-coastal in a subcontinuum i…

The even-cut decomposition conjecture. Every Peano continuum $X$ satisfying the even-cut condition admits, for every $\varepsilon>0$, a finite cover $\mathcal{U}$ of edge-disjoint…

The open Eulerianity conjecture. A Peano continuum $X$ is open Eulerian if and only if all but two vertices of $X_\sim$ have even degree.

The Eulerianity conjecture. A Peano continuum $X$ is Eulerian if and only if every edge cut of $X$ is even.

Crosscut extension conjecture. The following are equivalent:

Let $\mathcal{F}$ be an IFS on $\mathbb{R}^2$ and let $E$ be its attractor. Suppose that $E$ satisfies the open set condition and that every contraction in $\mathcal{F}$ involves n…

Pseudo-solenoid projective-homogeneity conjecture. For every $\epsilon>0$, there exists a homeomorphism $\varphi$ of $C$ onto itself such that, for every $x\in C$,

Pseudo-circle projective-homogeneity conjecture. For every $\epsilon>0$, there exists a local homeomorphism $\varphi$ of $C$ onto itself such that, for every $x\in C$,