On spaces with star kernel Menger

TL;DR

This paper introduces and studies star Menger spaces, exploring their properties, relationships with other topological properties, and behavior under various operations, while also addressing open questions in star selection theory.

Contribution

The paper defines star Menger spaces, investigates their properties, relationships, and behavior, and addresses open questions in star selection theory with new properties and questions.

Findings

Star Menger spaces are characterized and examples are provided.

The behavior of star Menger property under subspaces, products, and continuous images is analyzed.

Open questions in star selection theory are addressed and new properties are introduced.

Abstract

Given a topological property $\mathcal{P}$, a space $X$ is called star-$\mathcal{P}$ if for any open cover $\mathcal{U}$ of the space $X$, there exists a set $Y\subseteq X$ with property $\mathcal{P}$ such that $St(Y,\mathcal{U})=X$; the set $Y$ is called a star kernel of the cover $\mathcal{U}$. In this paper, we introduce and study spaces with star kernel Menger, that is, star Menger spaces. Some examples are given to show the relationship with some other related properties studied previously, and the behaviour of the star Menger property with respect to subspaces, products, continuous images and preimages are investigated. Additionally, some comments on the star selection theory are given. Particularly, some questions posed by Song within this theory are addressed. Finally, several new properties are introduced as well as some general questions on them are posed.