39 problems

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…

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

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 $X$ 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 $C_p(X)$ denote the space of continuous real-valued functions on a set of reals $X$, with the topology of pointwise convergence. Let $\mathfrak{rez}$ be the least cardinality o…

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…

Let $\mu$ be a singular cardinal of countable cofinality, and let $U(\mu)$ be the space of all uniform ultrafilters over $\mu$, with its usual topology whose basic open sets are…

Hechler-real scale conjecture. The scale induced by the Hechler reals is Fréchet.

Fréchet $\mathfrak{b}$-scale conjecture. There are Fréchet $\mathfrak{b}$-scales in the model constructed in this section.

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

An ultrafilter $U$ on $omega$ is a $p$-point if every countable subfamily of $U$ has a pseudointersection in $U$. A rapid $RK$-predecessor of $U$ is a rapid ultrafilter Tukey-relat…

Let $\kappa$ be an infinite cardinal, let $\omega$ denote the discrete space of natural numbers, and let $X$ be a connected manifold. Countable dense homogeneity extension conjectu…

The distinctness conjecture. If $\ne$, then

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…

Dow's conjecture. Souslin lines obtained by lexicographically ordering a Souslin tree are always functionally countable.

Fremlin's conjecture. The assertion is consistent with ZFC:

Nonexistence conjecture. $\operatorname{MA}+\neg\operatorname{CH}$ implies that there are no pseudo-Aronszajn compacta that are ccc.

A Dowker space is a normal topological space $\mathbb X$ whose product with the unit interval $\mathbb X\times[0,1]$ is not normal. Rudin's conjecture. There exists a Dowker space…

Near-coherence conjecture. If $\b5d=\b5u=\b5c$, then there exist uncountably many $\le_{\mathrm{RB}}$-incompatible $P$-points.

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…

Strong Ostaszewski space conjecture. $\diamondsuit$ implies that there is a strong $S$-space $X$ which is countably compact and perfectly normal, but not compact.

Let $A$ be a co-analytic subset of $\mathbb R$, and consider the Open Graph Axiom for co-analytic sets. Open Graph Axiom implication. The Open Graph Axiom for co-analytic sets impl…