Compactness

Abstract

A set is bounded if its points are not arbitrarily far from each other; but boundedness is not preserved by continuous functions. Totally bounded sets are special types of bounded sets that are preserved by uniformly continuous functions. Any sequence in such a set must have a convergent subsequence. A further strengthening of the definition leads to compact sets, which are preserved by continuous functions. Two equivalent formulations are given, one by the Heine-Borel theorem, as the complete and totally bounded subsets, and another by the Bolzano-Weierstraß theorem, when every sequence has a convergent subsequence in the subset. The chapter closes with the first non-trivial example of a complete metric space: the space $C(K)$ of continuous complex-valued functions on a compact metric space $K$. The Arzela- Ascoli theorem identifies its totally bounded subsets. The Stone-Weierstraß theorem states that the polynomials in $z$ and $\bar{z}$ are dense in $C(K)$ when $K\subseteq \mathbb {C}$.