9 problems

Gruenhage's 3-element basis conjecture. It is consistent with ZFC that $\mathcal{S}$ has a basis consisting of those three types of spaces.

A finite simplicial complex is the topological space obtained by gluing finitely many simplices along faces. A compact metrizable space is finitely self-similar if it arises from a…

Let $X$ be a first countable, compact space. Conjecture. The following are equivalent: 1. $C(X)$ is algebraically closed. 2. For every closed $Y\subseteq X$, every admissible…

A space $X$ is concentrated if it is uncountable and concentrated on some of its countable subsets, meaning that there is a countable set $D\subset X$ such that $X\setminus U$ is c…

Let $\mathcal{L}_{\vardiamond}$ be the modal language obtained by extending modal logic with the tangle operator $\vardiamond^\infty$, and let the topological $\mu$-calculus be the…

Square bracket polarized partition conjecture. If $(X,\tau)$ satisfies

Let $X$ and $Y$ be arc-connected spaces. For a nonzero cardinal number $\alpha$ and an ideal $\mathcal I$ on $X$, let $\mathfrak P_{\mathcal I}^\alpha(X)$ denote the corresponding…

First-countability conjecture. Every compact space with countable cs-character is first countable.

Conjecture on annihilator conditions. If condition (i) holds, then $X$ is an extremally $P$-space, and hence $C(X)$ is self-injective regular; if condition (ii) holds, then $X$ is…