Compactness with ideals

TL;DR

This paper introduces new notions of I compactness and I star compactness in topological spaces, overcoming previous issues with ideal convergence of subsequences, and explores their properties and distinctions from traditional compactness.

Contribution

It removes the obstacle in ideal convergence definitions and introduces I compactness and I star compactness, expanding the understanding of compactness via ideals in topology.

Findings

I compactness and I star compactness are distinct from classical compactness.

Involvement of nonthin subsequences differentiates these notions even in metric spaces.

The paper provides initial studies and properties of these new compactness concepts.

Abstract

One of the main obstacle to study compactness in topological spaces via ideals was the definition of ideal convergence of subsequences as in the existing literature according to which subsequence of an ideal convergent sequence may fail to be ideal convergent with respect to same ideal. This obstacle has been get removed in this article and notions of I compactness as well as I star compactness of topological spaces have been introduced and studied to some extent. Involvement of I nonthin subsequences in the definition of I and I star compactness make them different from compactness even in metric spaces.