A New Class of Locally Closed Sets and Locally Closed Continuous Functions in Generalized Topological Spaces

Abstract

ABSTRACT is closed under arbitrary union. The elements of In this paper the idea of -locally closed sets in generalized topological space is introduced and study some of their properties. We introduce the notion of (,’)-locally closed continuous functions on generalized topological space and investigate some of their characterizations. be generalized topological spaces. A map f: (X, Keywords -LC sets, (,’)–LC continuity, (,’)–LC irresoluteness. 1. INTRODUCTION Kuratowski and Sierpinski [5] considered the difference of two closed subsets of an n-dimensional Euclidean space. Implicit in their work is the notion of a locally closed subset of a topological space (X,τ). Following Bourbaki [3] we say that a subset of (X, τ) is locally closed in (X, τ) if it is the intersection of an open and closed subset of (X, τ). Stone [8] has used the term FG for a locally closed subset as the spaces that in every embedding are locally closed. The results of Borges [2] show that locally closed sets play an important role in the context of simple extensions. Blumberg [1] introduced the concept of a real valued function on Euclidean space being densely approached at a point of its domain. This notion was generalized in 1958 to general topological spaces by Ptak [7] who used the term nearly continuous function. The concepts of nearly continuous and nearly open functions are important in functional analysis especially in the context of open mapping and closed graph theorems. Csaszar [4] introduced the concepts of generalized neighborhood systems and generalized topological spaces. And also introduced the concepts of continuous functions and associated interior and closure operators on generalized neighborhood systems and generalized topological spaces. In particular, it has been investigated the characterizations for the generalized continuous function by using a closure operator defined on generalized neighborhood systems. In this paper we introduce the notion of -locally closed sets which are denoted by -LC sets and study some of the fundamental properties of -LC sets in generalized topological spaces. Also we introduce the concept of (,’)-locally closed continuous functions on generalized topological spaces and investigate the results of these functions.