Lebesgue's Number Lemma

In topology, Lebesgue's number lemma, named after Henri Lebesgue, is a useful tool in the study of compact metric spaces. It states:

If the metric space (X, d) is compact and an open cover of X is given, then there exists a number δ > 0 such that every subset of X having diameter less than δ is contained in some member of the cover.

Such a number δ is called a Lebesgue number of this cover. The notion of a Lebesgue number itself is useful in other applications as well.