Combinatorial aspects of selective star covering properties in $\Psi$-spaces

TL;DR

This paper investigates the star covering properties of Isbell--Mrówka ($$-spaces) using combinatorial and forcing methods, providing new examples and complete answers to existing questions in the field.

Contribution

It introduces a new example of a $$-space that is star-Menger but not star-Hurewicz, and offers a complete characterization of these properties using PCF-theory and forcing.

Findings

Constructed a $$-space that is star-Menger but not star-Hurewicz.

Provided a complete answer to a question of Bonanzinga and Matveev.

Extended results to Pixley--Roy spaces and free abelian topological groups.

Abstract

Which Isbell--Mr\'owka spaces ($\Psi$-spaces) satisfy the star version of Menger's and Hurewicz's covering properties? Following Bonanzinga and Matveev, this question is considered here from a combinatorial point of view. An example of a $\Psi$-space that is (strongly) star-Menger but not star-Hurewicz is obtained. The PCF-theory function $\kappa\mapsto\cof([\kappa]^\alephes)$ is a key tool. Using the method of forcing, a complete answer to a question of Bonanzinga and Matveev is provided. The results also apply to the mentioned covering properties in the realm of Pixley--Roy spaces, to the extent of spaces with these properties, and to the character of free abelian topological groups over hemicompact $k$ spaces.