A Unified view of (Complete) Regularity and Certain Variants of (Complete) Regularity

Abstract

Regularity and complete regularity are important topological properties and several generalizations of them occur in the literature on separation axioms. The properties of certain of these variants of (complete) regularity are similar to those of (complete) regularity and their theories run, either in part or in the whole, parallel to the theory of (complete) regularity. All the more, analogies inherent in their definitions as well as the nature of results obtained in the process of their study suggest the need of formulating a coherent unified theory encompassing the theory of (complete) regularity and its generalizations. An attempt leading towards the fulfillment of this need constitutes the theme of the present paper.Section 2 is devoted to preliminaries and basic definitions. In Section 3 we devise a framework which leads to the formulation of a unified theorv of (complete) regularity, almost (complete) regularity, ([26,], [27], [28]), (complete) s-regularity [13], (functionally) Hausdorff spaces, R1-spaces [3], and others.