Studies and applications of semi open sets in abstract cellular complex

Abstract

Topology is one of the most important fields in Mathematics which studies the properties of spaces that are invariant under continuous deformation The topological notions like open closed frontier connectedness boundary etc are closely related to the field of Image Analysis To apply all these topological notions in the field of Image Analysis Rosenfeld introduced the notion of Digital Topology and he represented a digital image as a neighborhood graph whose nodes are pixels and whose edges are linking adjacent pixels to each other Though this representation is very much useful for the study of images it contains two paradoxes namely connectivity and boundary paradoxes To overcome these difficulties Kovalevsky initiated a new concept namely Abstract Cellular Complex and established that every Abstract Cellular Complex is isomorphic to Finite Topological Spaces with T0 Separation Axiom This motivates us to study various topological notions on Abstract Cellular Complex Levine introduced the concepts of semi open set and semi continuity in Topological spaces and studied some of their properties Mashhour et al introduced the notion of pre open sets and pre continuous functions in Topological spaces This thesis is mainly focuses on the concept of semi open sub complexes in Abstract cellular complex and investigates some of their basic properties through semi frontier semi interior and semi closure Further the concept of semi neighborhoods local semi neighborhood semi continuous and semi homeomorphism are introduced Kovalevsky s Chain code algorithm to trace the boundary of an object in an image is compared with Moore s Neighborhood algorithm Moreover the boundary tracing algorithm using semi subcomplex is proposed and the proposed algorithm is compared with Kovalevsky s chain code algorithm newline