6 - Topological Spaces with Special Properties

Abstract

This chapter discusses topological spaces with special properties. Topological spaces play an important role in a variety of applications if they exhibit certain special structural properties. Connectedness, separability, compactness, and completeness are the most important special properties. A topological space is connected if it cannot be represented as the union of two separated sets. Any singleton subset {x} of an arbitrary space is connected because it does not contain any proper subset, so that the condition on connectedness is vacuously fulfilled. Homotopy is an equivalence relation for the collection of paths with specified common beginning and endpoints. A connected space is said to be simply connected if all loops at all points are contractable. A connected space is called n-fold connected if each point has exactly n connections to itself, that is, if at each point, the loop system has n distinct equivalence classes. Separability plays an important role in the theory of Hilbert spaces.