On a variation of selective separability: S-separability
Debraj Chandra, Nur Alam, Dipika Roy
TL;DR
This paper defines and investigates $S$-separability, a selective separability notion that sits between $H$- and $M$-separability. It develops the basic theory, showing $S$-separability is preserved by dense/open subspaces and by certain images, but not by all continuous maps, and reveals nuanced product behavior with a complete compact-space characterization via $\pi w(X)=\omega$. The work links $S$-separability to $C_p(X)$ through selection principles, establishes sharp thresholds at $\mathfrak b$ and $\mathfrak d$ for countable subspaces, and introduces $L$-separability as a related notion with its own placement among separability properties. Together, these results deepen understanding of how selective separability interacts with topological constructions, function spaces, and cardinal characteristics, while outlining several open questions for ZFC and consistency frameworks.
Abstract
A space $X$ is M-separable (selectively separable) (Scheepers, 1999; 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 $\cup_{n\in \mathbb{N}} F_n$ is dense in $X$. In this paper, we introduce and study a strengthening of M-separability situated between H- and M-separability, which we call S-separability: 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 for each finite family $\mathcal F$ of nonempty open sets of $X$ some $n$ satisfies $U\cap F_n\neq\emptyset$ for all $U\in \mathcal F$.