15 problems

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…

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…

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

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 $X$ 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…

Product selection-principle theorem. If $Y$ has ${\sf S}_c(\mathcal{O},\mathcal{O})$ and ${\sf S}_{fin}(\Omega,\mathcal{O}^{gp})$, then $X\times Y$ has…