Further investigations on certain star selection principles

TL;DR

This paper explores star versions of classical selection principles like Menger, Hurewicz, and Rothberger, using advanced set-theoretic tools to uncover new properties and relationships in topology.

Contribution

It introduces new observations on star selection principles, compares different star properties, and provides diagrams and tables to clarify their interactions.

Findings

New insights into star Menger, Hurewicz, and Rothberger properties

Relationships between star selection principles and cardinal invariants

Implication diagrams illustrating property interactions

Abstract

We consider certain star versions of the Menger, Hurewicz and Rothberger properties. Few important observations concerning these properties are presented, which have not been investigated in earlier works. A variety of investigations is performed using Alster covers and critical cardinalities $\mathfrak{d}$, $\mathfrak{b}$ and ${\sf cov}(\mathcal{M})$. Our study explores further ramifications on the extent and Alexandroff duplicate. In the process we present investigations on the star versions of the Rothberger property and compare with similar prior observations of the star versions of the Menger and Hurewicz properties. We sketch few tables that interpret (mainly preservation-kind of) properties of the star selection principles obtained so far. We also present implication diagrams to explicate the interplay between the star selection principles.