Compactness

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-Weierstraßtheorem, 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-Weierstraßtheorem states that the polynomials in z and \(\bar {z}\) are dense in \(C(K)\) when \(K\subseteq \mathbb {C}\) .