Comparing weak versions of separability

TL;DR

This paper explores various weak forms of separability in topological spaces, constructing examples to distinguish between classes with sigma-discrete and meager dense sets, and addresses open questions in the field.

Contribution

It introduces new examples and clarifies differences among weak separability properties, answering several open questions in topology.

Findings

Different classes of spaces with sigma-discrete and meager dense sets are distinguished.

Constructed examples demonstrate the distinctions between weak separability properties.

Resolved open questions posed by Tkachuk, Hutchinson, and others.

Abstract

Our aim is to investigate spaces with sigma-discrete and meager dense sets, as well as selective versions of these properties. We construct numerous examples to point out the differences between these classes while answering questions of Tkachuk [30], Hutchinson [17] and the authors of [8].