26 problems

Hurewicz conjecture. A set of reals has the Hurewicz property $\mathsf{U}_{\mathrm{fin}}(\mathcal{O},\Gamma)$ if and only if it is a countable union of its compact subsets.

Scheepers Conjecture. A topological space is a wQN-space if and only if it satisfies $S_1(\Gamma,\Gamma)$.

Additivity conjecture.

Let $X$ be a zero-dimensional separable metrizable space. The selection property $\mathsf{Split}(\Lambda,\Lambda)$ means that every large open cover of $X$ can be partitioned into…

Let $X$ be a set of reals with strong measure zero, and suppose that $|X|<\mathfrak{b}$, where $\mathfrak{b}$ is the bounding number. A finite power of $X$ is a Cartesian product…

A topological space $X$ is a perfectly normal wQN-space if it is perfectly normal and has the wQN property; equivalently in this context, it is an $s_1$-space, meaning that…

Let $X\subseteq\mathbb{R}$, and let $\{\mathcal O_n\}_{n\in\mathbb{N}}$ be the relevant sequence of open-cover classes. Write $\mathrm{T}$ for the collection of open $\tau$-covers,…

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 $X$ be a topological space, and let $\mathcal F[X]$ denote its finite-set hyperspace. A space is discretely selective if it has the selection property referred to in the preced…

Let $X$ be a subset of Cantor space. Write $\sogg$ for the relevant selection property, and distinguish its restriction to clopen covers from its version for open covers. Scheepers…

Projectively Scheepers Diagram conjecture. The projectively Scheepers Diagram is $t$-invariant; that is, each projective selection property in the Scheepers Diagram is preserved un…

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

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

Winning-strategy conjecture. For each positive integer $n$, ONE has a winning strategy in

Let $X,Y\subset\mathbb{R}$ be gamma spaces, and let $\mathsf{CH}$ denote the continuum hypothesis. The product space $X\times Y$ is said to have the selection property…

Productive weak-property conjectures. If a space is productively Menger then it is productively weakly Menger. If a space is productively Hurewicz then it is productively weakly Hu…

Product-space conjecture. The space

Let $Y$ be a subspace of a space $X$. It is a relative $\gamma$-set in $X$ when the selection principle $\mathsf{S}_1(\Omega_X,\Gamma_Y)$ holds. In particular, relative $\gamma$-se…

Let $X$ be a topological space. The strongly star-Hurewicz property means that for every sequence $(\mathcal U_n:n\in\mathbb N)$ of open covers of $X$, there are finite sets…

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…

Let $(G,*)$ be a metrizable topological group, and write $G^2=G\times G$ for its square. The conjecture. There is a metrizable group $(G,*)$ which has the property … and $G^2$ has…

Let $(G,*)$ be a metrizable topological group, and let $G^2=G\times G$ denote its square. The conjecture. There is a metrizable Menger bounded group $(G,*)$ with property … such th…

A topological group is Menger when it satisfies the Menger selection property, equivalently ${\rm S}_{\rm fin}(\mathcal O,\mathcal O)$. The principles…

A topological group $(G,*)$ is Hurewicz bounded when it satisfies ${\rm S}_1(\Omega_{\rm nbd},\mathcal O^{\rm gp})$. The principle ${\rm S}_c(\mathcal O_{\rm nbd},\mathcal O)$ conc…

A topological group $(G,*)$ is Menger bounded when it satisfies the relevant neighborhood selection property, and ${rm S}c( obreaktext{mathcal O}{rm nbd},mathcal O)$ and…