Studies on minimal structures In topological spaces

Abstract

The most basic and traditional division of topology is point set newlinetopology which describes the compactness and connectedness of topological newlinespaces The extension of traditional closed sets to g closed sets semi open newlineopen sets and many more associated concepts introduces newlinethe weaker forms of open sets whose complement sets belonging to closed newlinesets Based on the earlier studies the research done On weakly generalized newlineclosed sets and the present study attempt to explore further properties of newlineweakly generalized closed maps Minimal structure is an interesting newlineinnovation where in weakly generalized closed sets are studied Union and newlineintersection properties of mg closed sets are studied and proved that union of newlinetwo mg closed sets is not mg closed but the intersection of two mg closed newlinesets is an mg closed set In the next phase of the study mg continuous newlinefunctions mg T2 space subminimal structure mg compact maps mgirresolute newlinemaps and mg homeomorphism are introduced and their properties newlineare also proved Weakly generalized closed sets in minimal structure mwgclosed newlineset mwg continuous functions are introduced and their basic newlineproperties are obtained Since the concepts of uncertainty in information and newlinecomplexity are explained by fuzzy sets the term fuzzy is used in all branches newlineof mathematics the present study attempting to explain some of the minimal newlinestructure concepts in connection with fuzzy sets Fuzzy minimal generalized newlineclosed sets fmg closed sets fuzzy minimal weakly continuous functions newlinefuzzy minimal generalized irresolute maps and fuzzy minimal generalized newlinehomeomorphisms are introduced and their properties are studied newline newline