A Study on Generalized Closed Sets and Their Properties in Topological Spaces

Abstract

The notion of generalized closed sets, first introduced by Levine, occupies a central place in the study of general topology, since it provides a natural bridge between classical closed sets and several weaker forms of closedness studied subsequently in the literature. In this paper, we present a systematic study of generalized closed sets (g-closed sets) and generalized open sets (g-open sets) in topological spaces. We recall the fundamental definitions, establish several characterizing theorems, examine the relationship of g-closed sets with other well known classes of sets such as semi-closed, pre-closed and α-closed sets, and investigate their behaviour under subspace topology and continuous mappings. Suitable examples are provided to justify that the inclusions among these classes of sets are, in general, strict. The paper concludes with a discussion of generalized continuity and some open problems for further research.