On certain weaker forms of the Scheepers property

TL;DR

This paper introduces and explores weaker variants of the Scheepers property in topology, analyzing their properties, relationships, and implications for cardinalities, with examples and open problems.

Contribution

It defines new weaker forms of the Scheepers property, investigates their properties, equivalences, and cardinality considerations, and examines their behavior under various topological constructions.

Findings

Almost Lindelöf spaces of size less than  are S.

Finite powers of a space being M or M imply S or S.

Alexandroff duplicate of a space has Scheepers iff it has S_k.

Abstract

We introduce the weaker forms of the Scheepers property, namely almost Scheepers (${\sf aS}$), weakly Scheepers in the sense of Sakai (${\sf wS}$) and weakly Scheepers in the sense of Ko\v{c}inac (${\sf wS_k}$). We explore many topological properties of the weaker forms of the Scheepers property and present few illustrative examples to make distinction between these spaces. Certain situations are considered when all the weaker forms are equivalent. We also make investigations on the weak variations as considered in this paper concerning cardinalities. In particular we observe that 1. If every finite power of a space $X$ is ${\sf aM}$ (respectively, ${\sf wM}$), then $X$ is ${\sf aS}$ (respectively, ${\sf wS}$). 2. Every almost Lindel\"{o}f space of cardinality less than $\mathfrak{d}$ is ${\sf aS}$. 3. Let $X$ be Lindel\"{o}f and $\kappa<\mathfrak d$. If $X$ is a union of $\kappa$ many ${\sf aH}$ (respectively, ${\sf wH}$, ${\sf wH_k}$) spaces, then $X$ is ${\sf aS}$ (respectively, ${\sf wS}$, ${\sf wS_k}$). 4. The Alexandroff duplicate $AD(X)$ of a space $X$ has the Scheepers property if and only if $AD(X)$ has the ${\sf wS_k}$ property. 5. If $AD(X)$ is ${\sf aS}$ (respectively, ${\sf wS}$), then $X$ is also ${\sf aS}$ (respectively, ${\sf wS}$). Besides, few observations on productively ${\sf aS}$, productively ${\sf wS}$ and productively ${\sf wS_k}$ spaces are presented. Some open problems are also given.