Analysis of Closed Sets in Topological Spaces

Abstract

The endeavor of the thesis is to introduce and analyse a new class of closed sets, namely,and#916;*-closed (Delta star closed) sets in topological spaces. Their association with other existing closed sets, properties and characterizations are analysed under various concepts such as continuity, newlineirresoluteness, homeomorphisms and locally closedness. The nature of this newly defined closed sets are discussed in different topological spaces, like semi regular space, Hausdorff space, almost weakly newlineHausdorff space, R1-topological space, Partition space and T1/2-partition space. Also four new separation axioms are established and their imperative properties are proved as applications of and#916;*-closed sets. The extensions of homeomorphisms, namely, and#916;*-homeomorphism and and#916;*C-homeomorphisms are introduced and their significance in topological spaces are proved in this newlinethesis. Moreover the group property of and#916;*C-homeomorphisms is analyzed. The preservation of composition of two and#916;*-continuous functions, and#916;*-irresolute functions, and#916;*-homeomorphisms and newlineand#916;*-locally continuous functions are analyzed. All the concepts studied in topological spaces are extended to bitopological spaces and the analysis has been done in a similar way.The concept of and#916;*-closedness which is stronger than gand#948;-closedness but weaker than newlineand#948;g*-closedness is established in the thesis. It is proved that the composition of two and#916;*-continuous newlinefunctions, and#916;*-irresolute functions, and#916;*-homeomorphisms and and#916;*-locally continuous functions is newlinenot preserved. But after changing the conditions in different ways, it is shown that the newlinecomposition of mapping is preserved. Further the family of all and#916;*C-homeomorphisms forms a newlinegroup under the composition of maps. Also several results are analysed under and#916;*-locally closed sets. The analysis of and#916;*-closed sets is extended to bitopological spaces.