On a variation of selective separability using ideals

TL;DR

This paper introduces and investigates an ideal-based variation of H-separability in topological spaces, expanding the understanding of selective separability properties using ideals.

Contribution

It defines and explores the properties of $ ext{I}$-H-separability, a new concept extending classical H-separability with ideal-based conditions.

Findings

Characterization of $ ext{I}$-H-separability in various spaces

Relations between $ ext{I}$-H-separability and other separation properties

Examples illustrating the behavior of $ ext{I}$-H-separability

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.