Generalized Isolated Points and Generalized Perfect Sets in Topological Spaces

Abstract

This paper introduces the concepts of generalized isolated points (g-isolated points) and generalized perfect sets (g-perfect sets) within the framework of topological spaces, using the notions of generalized open and generalized closed sets. In this context, the concept of generalized scattered spaces (g-scattered spaces) is also defined. A series of fundamental results is established to clarify various properties of these generalized topological notions. Furthermore, the study provides several characterizations and presents illustrative examples that reveal their structure and demonstrate their connections with certain classical topological notions.