A New form of Generalized Closed Sets in General Topology

Abstract

The endeavor of the thesis is to introduce the concept of and#61561;*and#945;-closed sets in topological and newlinebitopological spaces. Their association with other existing closed sets, properties and characterizations newlineare analyzed under various concepts such as continuity, irresoluteness and homeomorphisms in newlinetopological and bitopological spaces. It is proved that in topological space, finite union of and#61561;*and#945;-closed newlinesets is and#61561;*and#945;-closed and arbitrary intersection of and#61561;*and#945;-closed sets is and#61561;*and#945;-closed. In bitopological space newlinefinite union of (i, j)-and#61561;*and#945;-closed sets is (i, j)-and#61561;*and#945;-closed but intersection of two (i, j)-and#61561;*and#945;-closed sets newlineneed not be (i, j)-and#61561;*and#945;-closed. Five new spaces are constructed in topological spaces and their newlineproperties are analyzed. Four new spaces are introduced in bitopological spaces and their newlineinterrelations are derived. It is shown that, and#61561;*and#945;-continuity is not preserved under composition of newlinemaps. But by changing the topological spaces and#61561;*and#945;-continuity is preserved under composition. newlineThe extension of homeomorphisms namely, and#61561;*and#945;-homeomorphisms and newlineand#61561;*and#945; -homeomorphisms are introduced in topological and bitopological spaces.. The group property of newlineand#61561;*and#945; -homeomorphisms is analyzed. Quasi and#61561;*and#945;-open maps, Quasi and#61561;*and#945;-closed maps and and#61561;*and#945;-quotient newlinemaps are introduced in topological spaces. Further and#61561;*and#945;-compactness and and#61561;*and#945;-connectedness are also newlinediscussed in topological spaces. newlinei) Major objectives : newlineThe major objective is to find weaker forms of closed sets which will satisfy certain newlineconditions which are not generally satisfied by closed sets. newlineii) Hypothesis: - newlineiii) Methodology : newlineA new form of generalized closed sets called and#61561;*and#945;-closed sets is studied by the following methods newlineand#61656; Analytical comparison with existing closed sets. newlineand#61656; Proving with illustrations. newlineand#61656; Analysis of preservation of topological properties. newlineand#61656; Obtaining Characterization theorems. newlineiv) Findings: newlineThe thesis is to investigate a new form of generalized closed sets called and#61561;*and#945;-closed sets in newlinetopological and bitopological spaces. newlineThe preliminary definitions and notations used in the the