Editing Heine–Borel theorem (section)

== Generalization of the Heine-Borel theorem ==

In general metric spaces, we have the following theorem:

For a subset <math>S</math> of a metric space <math>(X, d)</math>, the following two statements are equivalent:
* <math>S</math> is compact,
* <math>S</math> is precompact<ref>A set <math>S</math> of a metric space <math>(X, d)</math> is called precompact (or sometimes "totally bounded"), if for any <math>\epsilon > 0</math> there is a finite covering of <math>S</math> by sets of diameter <math>< \epsilon</math>.</ref> and complete.<ref>A set <math>S</math> of a metric space <math>(X, d)</math> is called complete, if any [[Cauchy sequence#In a metric space|Cauchy sequence]] in <math>S</math> is convergent to a point in <math>S</math>.</ref>

The above follows directly from [[Jean Dieudonné]], theorem 3.16.1,<ref>Diedonnné, Jean (1969): Foundations of Modern Analysis, Volume 1, enlarged and corrected printing. Academic Press, New York, London, p.&nbsp;58</ref> which states:

For a metric space <math>(X, d)</math>, the following three conditions are equivalent:
* (a) <math>X</math> is compact;
* (b) any infinite sequence in <math>X</math> has at least a cluster value;<ref>A point <math>x\in X</math> is said to be a cluster value of an infinite sequence <math>(x_n)</math> of elements of <math>x_n \in X</math>, if there exists a subsequence <math>(x_{n_k})</math> such that <math>x = \lim_{k\to\infty}x_{n_k}</math>.</ref>
* (c) <math>X</math> is precompact and complete.