Various $S(n)$-closednesses in $S(n)$-spaces with examples

TL;DR

This paper explores various closure properties in $S(n)$-spaces, providing examples that clarify their relationships and addressing open problems in the context of Lindel"{o}f spaces.

Contribution

It constructs and analyzes examples illustrating the relationships among different $S(n)$-closure types and solves some open problems in the field.

Findings

Relationships between $S(n)$-closed and $S(n)$-$ heta$-closed spaces clarified.

Examples demonstrate distinctions in Lindel"{o}f spaces.

Some open problems by Dikranjan and Giuli are solved.

Abstract

In this paper we continue to study various types of closures in $S(n)$-spaces. The main results are related to the construction and illustration of examples that allow us to understand the relationship between $S(n)$-closed, $S(n)$-$\theta$-closed, weakly $S(n)$-closed and weakly $S(n)$-$\theta$-closed spaces for each $n\in \mathbb{N}$. The relation of these classes in Lindel\"{o}f spaces is shown. Some of the solved problems formulated by D. Dikranjan and E. Giuli are presented in the examples.