Completeness and Separability

Abstract

A sequence may appear to converge but fails to do so because its limit is ‘missing’ from the metric space. Such sequences are called ‘Cauchy’; these, and the related concept of asymptotic sequences, are defined in this chapter. Complete metric spaces are spaces in which Cauchy sequences converge. Every metric space can be ‘completed’; for example, $\mathbb {R}$ is the completion of ${\mathbb {Q}}$. Baire’s theorem shows that the resulting complete space may be quite larger than the original space. Uniformly continuous functions preserve Cauchy sequences and completeness. Examples include Lipschitz functions. The Banach fixed point theorem is a unifying principle about contractive Lipschitz maps that has applications in various mathematical fields. Separable spaces are metric spaces whose points can be approximated in a constructive manner.