# 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.

## Proof

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, .

## References

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