Some properties defined by relative versions of star-covering properties II

TL;DR

This paper explores relative star-covering properties like set star Menger and Hurewicz, establishing their position between countable compactness and countable extent, and provides examples and counterexamples related to their behavior.

Contribution

It introduces and analyzes new relative star-covering properties, proving their bounds, and constructs examples and counterexamples addressing open questions.

Findings

Set strong star Menger and Hurewicz properties lie between countable compactness and countable extent.

The extent of regular set star Menger or Hurewicz spaces cannot exceed the continuum.

Examples show these properties are not preserved under products with compact spaces.

Abstract

In this paper we consider some recent relative versions of Menger property called set strongly star Menger and set star Menger properties and the corresponding Hurewicz-type properties. In particular, using \cite {BMae}, we "easily" prove that the set strong star Menger and set strong star Hurewicz properties are between countable compactness and the property of having countable extent. Also we show that the extent of a regular set star Menger or a set star Hurewicz space cannot exceed $\frak c$. Moreover, we construct (1) a consistent example of a set star Menger (set star Hurewicz) space which is not set strongly star Menger (set strongly star Hurewicz) and show that (2) the product of a set star Menger (set star Hurewicz) space with a compact space need not be set star Menger (set star Hurewicz). In particular, (1) and (2) answer to some questions posed by Ko\v{c}inac, Konca and Singh in \cite{KKS-MS} and \cite{S-AM}.