Compactness and Connectedness

Abstract

Those subsets of a euclidean space Eⁿ which are both closed and bounded† will be of special importance to us. As examples we mention the surfaces described in Chapter 1 and the finite simplicial complexes which we shall construct in Chapter 6 in order to triangulate spaces. We shall show that one can characterize these subsets by a purely topological property, that is to say a property which involves only the topological structure of Eⁿ and makes no mention of the idea of distance. This property, when formulated for topological spaces in general, is called ‘compactness’.