SC*-Regular spaces and some functions

TL;DR

This paper introduces $SC^*$-regular spaces defined through $SC^*$-open sets and investigates their relation to classical regularity notions such as regular, almost, softly, weakly, $\alpha$-regular, and $\zeta$-regular spaces. It develops a broad framework of generalized set and function notions—$SC^*$-open/closed, $gSC^*$-closed, $SC^*g$-closed, almost $SC^*$-irresolute, and related closure/interior operators—and establishes equivalences and preservation theorems linking these concepts to $SC^*$-regularity. The manuscript further analyzes how $SC^*$-regularity behaves under mappings, subspaces, and compositions, presenting a suite of theorems (including several equivalence statements and composition rules) that unify generalized regularity with standard topological separation axioms. Collectively, the results advance a cohesive framework for generalized regularity in topology and provide tools for studying spaces with weaker separation axioms and their associated functional mappings.

Abstract

This paper introduces a novel class of topological spaces, termed SC*-regular spaces, which are defined using SC*-open sets. We explore their fundamental properties and examine their connections with existing regularity concepts, such as regular, almost, softly, weakly, alpha, zeta, and generalized-regular spaces respectively. Furthermore, we examined and define, analyze generalized SC*-closed sets and SC*-generalized closed functions, establishing key properties and preservation theorems. Several characterizations of SC*-regular spaces are also presented, providing new insights into generalized regularity in topology.