Variations of star selection principles on small spaces

TL;DR

This paper introduces new star selection principles for small spaces, unifying various properties and exploring their relationships, with applications to the extent of certain separable spaces and their cardinal invariants.

Contribution

It defines the notions of Star-$\sigma ext{-}\mathcal{K}$ and absolutely Star-$\sigma ext{-}\mathcal{K}$ spaces, unifying and extending results in star selection principles.

Findings

The extent of separable absolutely strongly star-Menger spaces is at most the dominating number $rak{d}$.

The extent of separable absolutely strongly star-Hurewicz spaces is at most the bounding number $rak{b}$.

Connections between star properties and classical cardinal invariants are established.

Abstract

In this paper, we introduce the notions of Star-$\sigma\mathcal{K}$ and absolutely Star-$\sigma\mathcal{K}$ spaces which allow us to unify results among several properties in the theory of star selection principles on small spaces. In particular, results on star selective versions of the Menger, Hurewicz and Rothberger properties and selective versions of property $(a)$ regarding the size of the space. Connections to other well-known star properties are mentioned. Furthermore, the absolute and selective version of the neighbourhood star selection principle are introduced. As an application, it is obtained that the extent of a separable absolutely strongly star-Menger (absolutely strongly star-Hurewicz) space is at most the dominating number $\mathfrak{d}$ (the bounding number $\mathfrak{b}$).