A study on delta s g star Closed Sets in Topological Spaces

Abstract

The endeavor of the thesis is to introduce the concept of and#948;sg*-closed (and#948; semi generalized star newlineclosed) sets in topological spaces. Their association with other existing closed sets, properties and newlinecharacterizations are analyzed under various concepts such as continuity, irresoluteness, newlinehomeomorphisms, locally closed and somewhat continuity. The nature of this newly defined closed newlinesets are discussed in different topological spaces, like semi regular space, semi weakly Hausdorff newlinespace, R1-topological space and T1/2-partition space. Seven new separation axioms are constructed and newlinetheir interrelations with existing spaces are analyzed. newlineThe extension of homeomorphisms namely, and#948;sg*-homeomorphisms and newlineand#948;sg*C-homeomorphisms are introduced. The interrelations of these newly introduced functions with newlinevarious functions are analyzed and the dependence elucidated. Furthermore, and#948;sg*-compactness and newlineconnectedness are discussed. In addition, the group property of and#948;sg*C-homeomorphisms is analyzed. newlineIt is shown that, the composition of two and#948;sg*-continuous functions, and#948;sg*-closed functions and newlineand#948;sg*-homeomorphisms are not preserved. But after modifying the conditions in different ways the newlinecomposition of functions are preserved. Also several results are obtained under and#948;sg*-locally closed newlinesets. newlineVarious types of somewhat functions related to somewhat and#61540;sg*-continuity, somewhat almost newlineand#61540;sg*-continuity, somewhat and#61540;sg*-irresoluteness, somewhat and#61540;sg*-openness and somewhat almost newlineand#61540;sg*-openness are defined. Many properties and characterizations of newly defined somewhat newlinefunctions are derived and presented. 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 : newlineThe study of and#61540;sg*-closed sets has been done by the following methods. newlineand#61656; Analytical method of comparing and#61540;sg*-closed sets with other existing closed sets. newlineand#61656; Producing counter examples wherever necessary. newlineand#61656; Analysis of preservation of topological