21 problems

Let $X$ be a compact topological space, and suppose every closed subset of $X$ is a $G_\delta$ set, meaning a countable intersection of open subsets of $X$. Fremlin's two-to-one co…

A basis of topological spaces means that every space in the class has one of the listed spaces as the relevant basis representative. Gruenhage's three-element basis conjecture. The…

Let $X$ be a regular Hausdorff space. A space is hereditarily separable if every subspace is separable, and hereditarily Lindelöf if every subspace is Lindelöf. Hajnal–Juhász basis…

Non-determinacy problem. The game $G_{\omega}(Y,\mathbb{A})$ is not determined.

Equivalence conjecture for $\sigma E$.

Equality conjecture. If $X$ is almost zero-dimensional, then

The strongly star-Scheepers game conjecture. If $X$ is strongly star-Scheepers, then ONE does not have a winning strategy in the game…

Expected inclusion.

Let $X$ be a metrizable space, and let $F_4(X)$ denote the corresponding free topological group of length at most four. Suppose that the set of all non-isolated points of $X$ is co…

For a topological space $X$, let $\operatorname{Compl}(X)$ be its set of attainable complexities. Classification conjecture. Exactly one of the following alternatives holds: … More…

For a space $X$, let $\operatorname{Compl}(X)$ denote its set of attainable complexities. A space is $\mathcal K$-analytic if it has the standard $\mathcal K$-analyticity property…

Let $d,s\geq 1$. Suppose that $L_1,\ldots,L_d$ are linearly ordered compact spaces and there is a continuous surjection … where $K_1,\ldots,K_{d+s}$ are infinite compact spaces. Ma…

One-point connectification conjecture. Let $i=3$. A locally connected $T_i$-space has a $T_i$ one-point connectification if and only if it has no compact (or almost compact) compon…

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…

A rank $n$-diagonal is a diagonal satisfying the rank-$n$ condition described in the surrounding theory. Diagonal-rank separation conjecture. For every natural number $n$ there is…

Weak-space characterization conjecture. $C_c(X)$ is a weakly $\aleph$-space if and only if $C_c(X)$ is a weakly $\aleph_0$-space if and only if $X$ is a countable $\aleph_0$-space.

Let $(X,\tau)$ be a topological space, and let $PR(X)=[X]^{<\omega}$ be the set of finite subsets of $X$ equipped with the Pixley–Roy topology, whose basic open sets are … for…

A complete admissible metric $d$ for a $\sigma$-compact, locally compact space $X$ is a metric inducing the topology of $X$ and complete as a metric space. A Heine–Borel metric is…

Product-space conjecture. The space

Characterization and cardinal-invariant conjecture. begin{enumerate} item A compact space $X$ is D-separable if and only if $X$ has a $\sigma$-disjoint $\pi$-base. item For every c…

Let $X$ be a metrizable space. The Menger–screenability conjecture asserts that there is such an $X$ with the Menger property and … which does not have the Haver property in some m…