A Unified Theory on Some Basic Topological Concepts

TL;DR

This paper proposes a unified theoretical framework for various topological concepts, including compactness and closure, extending previous unifications in general and fuzzy topological spaces.

Contribution

It introduces the concept of $\,\phi_{1,2}$-compactness and explores its relationships with filters and closure operators, advancing the unification of topological properties.

Findings

Defined $\,\phi_{1,2}$-compactness and analyzed its properties.

Established connections between $\,\phi_{1,2}$-compactness, filters, and closure operators.

Extended previous unification efforts in topological space concepts.

Abstract

Several mathematicians, including myself, have studied some unifications in general topological spaces as well as in fuzzy topological spaces. For instance in our earlier works, using operations on topological spaces, we have tried to unify some concepts similar to continuity, openness, closedness of functions, compactness, filter convergence, closedness of graphs, countable compactness and Lindelof property. In this article, to obtain further unifications, we will study $\phi_{1,2}$-compactness and relations between $\phi_{1,2}$-compactness, filters and $\phi_{1,2}$% -closure operator.