A Study on Some Generalizations of Delta closed sets in Topological Spaces

Abstract

Topology is a widely studied area of mathematics emerged through the works of the great Mathematician Henri Poincare in the 19th century. Topology developed as a field of study out of geometry and set theory, through analysis of such concepts as space, dimension and transformation. It is the study of continuity and connectivity. The topological structures are modeled suitably in the fields of computer graphics, pattern recognition, artificial intelligence, data mining, rough set theory, information systems, quantum physics etc. newlineThe notion of open sets is the powerful tool for defining a topological space. Levine (1960) introduced and studied the concepts of semi-open sets in topological spaces, as a weaker form of open sets. It is found from literature that during recent years many topologists are interested in the study of generalized types of closed sets. The study of generalized closed sets was initiated by Levine (1970) in order to extend some important properties of closed sets to a larger family of sets. The productivity and fruitfulness of the notion of generalized closed sets motivated the mathematicians to introduce weaker and stronger forms of generalized closedness for the past four decades. With the aid of g-open sets, they introduced, investigated and modified continuous functions which are the core concept of topology. Detailed study in this regard by many investigators has enriched the field of generalized closed sets to a considerable extent. newlineOpen maps and closed maps are very useful in topological spaces. The concept of homeomorphisms plays an important role in topological spaces. For researchers on various closed sets, the study is not complete without extending their definitions to open (closed) maps and homeomorphisms. Malghan (1982) introduced the concept of generalized closed maps in topological spaces. Maki et al. (1991) introduced generalized homeomorphisms and gc-homeomorphisms and studied their properties. newlineJ.C.Kelly (1963) introduced the idea of bitopological spaces and thereafter the th