Advances in General Topology and Its Application

Abstract

The mathematical study of continuity, connectedness and related phenomena in a broad context might be called generic topology. The real line, Euclidean spaces (including infinite dimensions), and function spaces are some of the places where these notions initially emerge. First, we must discover an abstract environment in which we can articulate findings of continuity and related notions (convergence, compactness, connectedness, and so forth) that occur in these more concrete situations. Frechet and Hausdor laid the foundations for general topology in the early 1900s. General topology is frequently referred to as point-set topology since it is based on the idea of sets. On the other side of the spectrum is algebraic topology, which applies abstract algebraic concepts to the study of de ne algebraic invariants of spaces; and, on the other hand, differential topology examines topological spaces with extra structure in order to study differentiability (basic general topology only generalizes the notion of continuous functions, not the notion of differentiable function). Since Bendse and Kikuchi's groundbreaking publication on topology optimization in 1988, topology optimization has evolved tremendously. "density," "level set," "topological derivative," 'phase field', and a slew of other terms are now being used to describe the notion. An overview, comparison, and critical analysis of the various techniques, their strengths, shortcomings, similarities and dissimilarities are presented in this work.