13 problems

Let $X$ be a topological space. Write {\sf S}_1(\{\mathcal O_n\}_{n\in\mathbb{N}),\mathrm{T}) for the property that, from every sequence of covers $\mathcal O_n$ of the relevant…

Cardinality and Ramsey-ultrafilter conjecture. For every ultrafilter $\boldsymbol u$,

Compactness conjecture. Every Nyikos inverse topological semigroup is compact.

Let $K$ be a positive natural number, and let $c:[\mathbb R]^2 \rightarrow K$ be a colouring of pairs of reals. Galvin's conjecture. There is a set of reals $Y$ homeomorphic to…

Todorčević's conjecture. It is relatively consistent with $\mathsf{ZFC}$ that if $X$ is a regular $\mathsf{HL}$ space, then $\mathsf{OGA}(X)$ holds.

Todorčević's conjecture. It is relatively consistent with $\mathsf{ZFC}$ that if $X$ is a regular space with no uncountable discrete subspace, then $\mathsf{OGA}(X)$ holds.

Let $\kappa$ be a singular cardinal, and let ${}^{<\kappa}\square\,2^\kappa$ denote the corresponding middle box product space. The paracompactness conjecture for singular cardinal…

Let $K$ be a nonempty Valdivia compact space satisfying the countable chain condition (ccc). Valdivia compacta ccc dichotomy conjecture. Either $K$ has a $G_\delta$ point or $K$ ad…

Let $X$ be a locally compact, non-compact, separable metrizable space, and let $X^*$ denote its Stone–Čech remainder. An autohomeomorphism of $X^*$ is a homeomorphism from $X^*$ to…

Let $X$ and $Y$ be locally compact Polish spaces. For a locally compact space $X$, write $X^* = \beta X\setminus X$ for its Stone–Čech remainder, and call a homeomorphism between s…

Let $\mathcal{A}$ be a separable, nonunital C-algebra, and let its corona be the quotient $\mathcal{Q}(\mathcal{A})=\mathcal{M}(\mathcal{A})/\mathcal{A}$ of its multiplier algebra…

Let $\mathcal{A}$ be a separable, nonunital C-algebra, and let its corona be the quotient $\mathcal{Q}(\mathcal{A})=\mathcal{M}(\mathcal{A})/\mathcal{A}$ of its multiplier algebra…

PFA(S)[S] conjecture. Under $\mathrm{PFA}(S)[S]$, every first-countable perfect pre-image of $\omega_1$ includes a copy of $\omega_1$.