Between Closed Sets And -Closed Sets

Abstract

Topology, as a branch of mathematics, deals extensively with the study of spaces and their properties under continuous transformations. Central to this study are concepts like closed sets, which encapsulate the notion of completeness and limit points within a space. Recently, the notion of -closed sets has emerged, offering a refined understanding of convergence in weak topologies. This paper explores the intricate relationship between closed sets and -closed sets within the framework of topology. It delves into the fundamental characteristics that distinguish these two types of sets and investigates their implications for the convergence of sequences and continuity within topological spaces. Through a comprehensive analysis, this paper elucidates the conditions under which a set can be considered both closed and -​-closed, shedding light on the subtle nuances of convergence and completeness. Additionally, it provides examples and counterexamples to illustrate the distinctions between these concepts, offering insights into the behaviour of sequences and their limit points. By bridging the gap between closed sets and -​-closed sets, this paper contributes to a deeper understanding of convergence properties in topological spaces. It serves as a valuable resource for researchers and practitioners seeking to explore the intricate relationships between different types of sets and their implications for the study of continuous transformations and spatial structures in topology