Complete Hausdorffness and Complete Regularity on Supra Topological Spaces

T M Al-Shami,Dan Huang

doi:10.1155/2021/5517702

Abstract

The supra topological topic is of great importance in preserving some topological properties under conditions weaker than topology and constructing a suitable framework to describe many real-life problems. Herein, we introduce the version of complete Hausdorffness and complete regularity on supra topological spaces and discuss their fundamental properties. We show the relationships between them with the help of examples. In general, we study them in terms of hereditary and topological properties and prove that they are closed under the finite product space. One of the issues we are interested in is showing the easiness and diversity of constructing examples that satisfy supra T i spaces compared with their counterparts on general topology.

Highlights

Introduction and PreliminariesTopological space has been generalized in many manners
The third one was established by defining a topology using one of the generalizations of crisp sets such as fuzzy topology [7] and soft topology [8]
Mashhour et al [2] introduced the supra topology concept by deleting only the intersection condition. They studied the concepts of interior and closure operators, continuity, and separation axioms on supra topology

Summary

Introduction and Preliminaries

Topological space has been generalized in many manners. They can be classified into three main types; the first one was obtained by strengthening or weakening the conditions of a topology such as Alexendroff topology [1], supra topology [2], and generalized topology [3]. Some published articles demonstrated that many topological findings are still true on supra topologies and illustrated that some of them are invalid like the distributive property of the closure (resp., interior) operator for the union (resp., intersection) between two sets and a compact set in a Hausdorff space is closed. Our goal in this article is to complete separation axioms reported in [19] by introducing the concepts of supra completely Hausdorff and supra completely regular spaces. We elaborate their master properties and demonstrate the relationships between them by some illustrative examples.

Supra Completely Hausdorff Spaces

Supra Completely Regular Spaces

Conclusion