A study on generalized ab closed sets in topological spaces bitopological spaces and fuzzy topological spaces

Abstract

Topology is the main branch of mathematics and it lays the newlinefoundation for the future study in analysis geometry and algebraic topology newlineThe term topology can be formally defined as the qualitative properties of newlinecertain objects called sets mappings and topological spaces The objects that newlineare invariant under certain kind of transformations are called continuous newlinemaps and those under a certain kind of equivalence are called newlinehomeomorphism The concept of topological space grew out of the study of newlinethe real line and Euclidean spaces and the study of continuous functions on newlinethese spaces The present research is done under the basis of the earlier and newlinetraditional studies It is a brief monograph which explores the new class of newlinegeneralized bclosed sets which investigate the intersection of two newlinegeneralized bclosed sets and the union of two generalized bclosed sets It newlinealso discusses its other properties newlineThe study of the separation axioms continuous function closed newlinemaps open maps and homeomorphism are also presented as a part of the newlinepresent thesis The main class of generalized bhomeomorphism which newlineforms the subclass of generalized bhomeomorphism is determined newlineprecisely and the class of map is closed under the composition of maps The newlinecomposition of two generalized bhomemorphism have been discussed It is newlineproved that the set of all the homeomorphism of generalized bchomeomorphism newlineforms a group under the binary operation of the newlinecomposition of maps newlineIn the present research some of the concepts and results of the newlineclassical topological spaces in the field of bitopology are expanded newlineMoreover innovative and informative study of bitopology is presented via newlinedefinitions illustrations and exemplifications It shows the relationship newlinebetween the generalized closed sets and bitopological generalized bclosed newlinesets It explains the interior closure and generalized bregular in the newlinebitopological spaces The concept of closed graph is discussed with the newlinebitopological spaces newline newline