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:
Such a number 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.
If any one of the 's equals then any will serve as a Lebesgue number.
Otherwise for each, let, note that is not empty, and define a function by.
Since is continuous on a compact set, it attains a minimum.
The key observation is that, since every is contained in some, the extreme value theorem shows. Now we can verify that this is the desired Lebesgue number.
If is a subset of of diameter less than, then there exists such that, where denotes the ball of radius centered at . Since there must exist at least one such that. But this means that and so, in particular,.