On a variation of selective separability: S-separability

TL;DR

This paper introduces S-separability, a new property of topological spaces that strengthens M-separability by ensuring finite subsets intersect all members of finite families of open sets, bridging H- and M-separability.

Contribution

The paper defines and investigates S-separability, a novel topological property that refines existing notions of selective separability, expanding the understanding of space separability conditions.

Findings

S-separability is strictly stronger than M-separability.

S-separability is weaker than H-separability.

The property has implications for the structure of dense subspaces.

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 $U\in \mathcal F$.