Iterations and unions of star selection properties on topological spaces

TL;DR

This paper explores how small unions of topological spaces with star selection properties behave, investigates their iterative properties, and examines their behavior on specific spaces, revealing new distinctions among these properties.

Contribution

It provides new results on unions and iterations of star selection properties, and demonstrates the existence of a normal star-Menger space that is not strongly star-Menger.

Findings

Small unions of spaces can possess certain star selection properties.

Iterated properties can differ from weaker variants like paracompactness.

Existence of a normal star-Menger space that is not strongly star-Menger under certain conditions.

Abstract

In this paper, we investigate what selection principles properties are possessed by small (with respect to the bounding and dominating numbers) unions of spaces with certain (star) selection principles.. Furthermore, we give several results about iterations of these properties and weaker properties than paracompactness. In addition, we study the behaviour of these iterated properties on $\Psi$-spaces. Finally, we show that, consistently, there is a normal star-Menger space that is not strongly star-Menger.