Studies on some algebraic structures in abstract cellular complex

Abstract

Topology, a well-known area of Mathematics, is a study of newlinerelationship between two spaces, particularly the geometric structures with their newlineproperties, deformations, and mapping between them. It has recently become newlinean important area of applied mathematics. The study of basis is useful to define newlinethe topological space of a set X, from the smaller collection of a subset X. Each newlinepoint x in X is enclosed in a family of subsets of topological space X, called its newlineneighbourhoods. newlineThe studies of connectedness and path connectedness are useful to newlinecharacterize the different topological spaces. For example, removing a point newlinefrom the plane R×R leaves a connected space remaining whereas removing newlinea point from the real line R does not. It is insufficient in the case of torus newlineand sphere while using usual invariants of topology. Further, identifying newlinesimilar topological spaces using homeomorphism is a challenging task for newlinetopologists. Hence, topologists introduced the concept of fundamental groups newlinein topological spaces by initiating the notions of homotopy in topological spaces newlineand by establishing that the two topological spaces are homeomorphic if their newlinecorresponding fundamental groups are isomorphic to each other. Because newlinehomotopy is concerned about the classification of geometric regions by studying newlinethe paths which are drawn in the region. Moreover, homotopy gives information newlineabout the basic structure or holes of the space. newlineMany topological properties like connected, neighbourhood, closure, newlineinterior, boundary, etc., are used in digital image analysis. newline