A study on some open sets via regular open sets in topological space and bitopological space

Abstract

This thesis undertakes an in-depth investigation of some open sets in both topological and bitopological spaces by using regular open sets. The limitations of regular open set that the arbitrary intersections of regular open sets need not be regular open motivated to introduces the concept of Ir -open sets, defined as intersections of regular open sets. The thesis define Ir -closure, Ir -interior, and Ir -cluster points and presents fundamental prop- erties, theorems, and equivalent characterizations. The work systematically introduced and explored the generalized separation axioms ( Ir -T0 , Ir -T1 , Ir -T2 ), Ir -regularity, and Ir - normality, along with their starred counterparts, Ir -connectedness, and Ir -compactness. This thesis extends the framework to define Ir∗ -closed sets and investigates their proper- ties, characterizations, and behavior under set operations. The Ir -cover of a set is defined and used to construct a supra-topology τIr , within which several characterizations of sep- aration axioms are studied. It further examines function classes including Ir -continuous, Ir -irresolute, and Ir -open functions, along with highlighting their role in characterizing topological properties. The study advances into the notions of minimal and maximal reg- ular open sets and introduces somewhat mr-continuous and somewhat mr-open functions with their associated topologies. In the context of bitopological spaces, the thesis de-fines pairwise minimal and maximal regular open sets, proposes new separation axioms, spaces and investigates various continuity classes such as (i, j)-almost continuous, (i, j)- almost completely continuous, etc. Further, p-Min and p-Max r-continuous maps are defined, and their interrelations with existing function classes are studied, enhancing the structural understanding of bitopological continuity. Overall, this thesis provides a comprehensive and unified framework that not only generalizes classical topological concepts but also enriches the theoretical landscape of topological and bitopological spaces.