Compactness

Abstract

The concept of compactness is undoubtedly the most important idea in topology. We have seen the importance of closed and bounded subsets of a Euclidean space ${\mathbb {R}}^n$ in real analysis; these sets, in topological parlance, are “compact”. This term has been used to describe several (related) properties of topological spaces before its current definition was accepted as the most satisfactory. It was initially coined to describe the property of a metric space in which every infinite subset has a limit point. However, this sense of compactness failed to give some desirable theorems for topological spaces, especially the invariance under the formation of topological products.