A Generalization of Delta Closed Sets in Topological Spaces and Applications

Abstract

The aim of this work is to establish and study the properties of the notion of and#948; newlineg and#955; -closed newlinesets in topological spaces and fuzzy topological spaces. This is followed by the application of newlinethis notion in image processing in order to check for the efficiency of this notion in real life newlinescenario. newlinei) Major objectives : newlineand#61623; To establish the notion of and#948; newlineg and#955; -closed sets in topological and fuzzy topological newlinespaces. newlineand#61623; To study various notions related to and#948; newlineg and#955; -closed sets. newlineand#61623; To work on the applications of and#948; newlineg and#955; -closed sets. newlineii) Methodology : newlineThe research has been done by the following methods. newlineo Analytical method of comparing and#948; newlineg and#955; -closed sets with existing closed sets. newlineo Producing Counter examples wherever necessary to substantiate the result. newlineo Analysis of preservation of topological properties by and#948; newlineg and#955; -closed sets. newlineo Obtaining Characterization theorems. newlineo Apply the developed theory to suitable real life scenario and test its efficiency. newlineiii) Findings: newlineIn this thesis, the study of the notion and#948; newlineg and#955; -closed sets is presented. This notion is properly newlineplaced between and#948;-closed sets and and#948;g*-closed sets. It is the nearest weaker form of and#948;-closed sets. newlineand#948; newlineg and#955; -closed sets are much stronger than many existing generalizations of and#948;-closed sets. The newlinereverse implications of the dependence relationship, which do not hold good, are substantiated newlineby counter examples. newlineThe characterizations of and#948; newlineg and#955; -closed sets in semi-regular space, almost weakly newlineHausdorff space, T3/4-spaces are also derived. Properties of various notions related to and#948; newlineg and#955; -closed newlinesets namely and#948; newlineg and#955; -neighborhood, and#948; newlineg and#955; -limit point, and#948; newlineg and#955; -derived set, and#948; newlineg and#955; -frontier, and#948; newlineg and#955; -boundary, newlineand#948; newlineg and#955; -exterior and and#948; newlineg and#955; -saturated set are studied. and#948; newlineg and#955; -open sets are characterized using the concept of newlinegrill in topological spaces. newlineAs an application of and#948; newlineg and#955; -closed sets, five new spaces are constructed and their newlineinterrelations with existing spaces and among themselves are analyzed. It is shown that the newlinecomposition of two and#948; newlineg and#955; -continuous functions is not preserved. But after c