16 problems

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.

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

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 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 $x\in C$,

Let $K_{\mathrm{addr}}$ be the addresses odometer, identified with the Cantor set of connected components of $K_\infty$. A subset $B\subset K_{\mathrm{addr}}\times[0,1]$ is require…

Let $X$ be a totally regular continuum, meaning a continuum in which every two points can be joined by an arc of finite length. Write $I^{\operatorname{ED}}(X)$ for the infimum of…

Let $X$ be a Peano continuum, that is, a compact, connected, locally connected metrizable space. For a compatible metric $d$ on $X$, let $\hat{H}_d$ denote the associated metric-de…

Let $X$ be a compact, locally connected, metrizable space that is locally embeddable in $S^2$. A simple set of circles in $X$ is a set $S$ of circles with the simplicity property u…