7 problems

Let $X$ be a Hausdorff topological space, and let $(X_n)_{n\in\omega}$ be a superskeleton of $X$. For every $n<m$ in $\omega$, consider the subset $X_m$ of $X_n$ and the complement…

An $n$-dimensional Nöbeling space is a subset of $\mathbb{R}^{2n+1}$ consisting of points with at most $n$ rational coordinates. Nöbeling-space analogue conjecture. In the 1980s it…

Let the Hilbert cube be $mathbb I^\omega$, and let $mathcal Z_n$ denote the family of $n$-dimensional $Z$-sets, for $n\leq\omega$. A subset is $mathcal Z_n$-absorptive if it has th…

Let $\ell^2_+$ be the positive cone of the separable Hilbert space $\ell^2$. A subset $B\subset\ell^2_+$ is $\mathbb B$-convex if, for all $x,y\in B$ and $t\in[0,1]$, one has…

Let $\mathcal H^u(\mathbb R^n)_b$ denote the group of bounded uniform homeomorphisms of Euclidean space $\mathbb R^n$, equipped with the sup-metric, and let $\ell_\infty$ be the Ba…

Hurewicz-set translate conjecture. If $H\subset {\mathbb R}^{\mathbb N}$ is a Hurewicz set, then for each countable dimensional set $C\subset {\mathbb R}^{\mathbb N}$,

For an infinite cardinal $\tau$, let $\operatorname{cc}(\mathbb R^\tau)$ denote the hyperspace of compact subsets of the product space $\mathbb R^\tau$. Hyperspace homeomorphism co…