Star versions of Hurewicz spaces

TL;DR

This paper introduces and studies the set star Hurewicz property in topological spaces, exploring its relationships with related properties and analyzing the implications for the structure of such spaces.

Contribution

It defines the set star Hurewicz property, investigates its connections with other covering properties, and examines the topological characteristics of spaces possessing these properties.

Findings

Relationships among set star Hurewicz, set strongly star Hurewicz, and other covering properties established.

Topological properties of spaces with these covering properties analyzed.

New insights into the structure of spaces with star covering properties obtained.

Abstract

A space $X$ is said to have the set star Hurewicz property if for each nonempty subset $A$ of $X$ and each sequence $(\mathcal{U}_n: n \in \mathbb{N})$ of sets open in $X$ such that for each $n\in \mathbb N$, $\overline{A} \subset \cup \mathcal{U}_n$, there is a sequence $(\mathcal{V}_n: n \in \mathbb{N})$ such that for each $n \in \mathbb{N}$, $\mathcal{V}_n$ is a finite subset of $\mathcal{U}_n$ and for each $x \in A$, $x \in {\rm St}(\cup \mathcal{V}_n, \mathcal{U}_n)$ for all but finitely many $n$. In this paper, we investigate the relationships among set star Hurewicz, set strongly star Hurewicz and other related covering properties and study the topological properties of these topological spaces.