On a variation of selective separability using ideals
Debraj Chandra, Nur Alam, Dipika Roy
TL;DR
This paper defines $\mathcal{I}$-H-separability, an ideal-variant of $H$-separability, and situates it between $H$-separability and $M$-separability with invariance under open maps and subspaces. It proves that if $\mathcal{I}_1 \le_{1-1} \mathcal{I}_2$, then $\mathcal{I}_1$-H-separability implies $\mathcal{I}_2$-H-separability, and that $\mathcal{I}$-H-separability can be strictly weaker than $H$-separability in general. The work introduces the cardinal $\mathfrak{b}(\mathcal{I})$ and derives results for Cantor cubes $2^\kappa$, including that countable subspaces of $2^\kappa$ are $\mathcal{I}$-H-separable when $\kappa < \mathfrak{b}(\mathcal{I})$, and that there exist countable dense non-\mathcal{I}-H-separable subspaces of $2^{\mathfrak{b}(\mathcal{I})}$. It connects these ideas to function spaces by characterizing $\mathcal{I}$-H-separability of $C_p(X)$ via $\mathcal{I}$-WFS and $\mathcal{I}$-WFSD, and by relating them to the property that every finite power of $X$ is $\mathcal{I}$-Hurewicz, with consequences for separability and weight. Overall it advances the theory of ideal based refinements of selective separability and raises questions about invariance and consistency in ZFC.
Abstract
A space $X$ is H-separable (Bella et al., 2009) if for every sequence $(Y_n)$ of dense subspaces of $X$ there exists a sequence $(F_n)$ such that for each $n$ $F_n$ is a finite subset of $Y_n$ and every nonempty open set of $X$ intersects $F_n$ for all but finitely many $n$. In this paper, we introduce and study an ideal variant of H-separability, called $\mathcal{I}$-H-separability.