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.
Let be an open cover of . Since is compact we can extract a finite subcover .
For each , let and define a function by .
Since is continuous on a compact set, it attains a minimum . If is a subset of of diameter less than , then there exist such that , where denotes the radius ball centered at (namely, one can choose as any point in ). Since there must exist at least one such that . But this means that and so, in particular, .