Lebesgue's number lemma

In topology, the Lebesgue covering lemma is a useful tool in the study of compact metric spaces.

Given an open cover of a compact metric space, a Lebesgue's number of the cover is a number ${\displaystyle \delta >0}$ such that every subset of ${\displaystyle X}$ having diameter less than ${\displaystyle \delta }$ is contained in some member of the cover.

The existence of Lebesgue's numbers for compact metric spaces is given by the Lebesgue's covering lemma:

If the metric space ${\displaystyle (X,d)}$ is compact and an open cover of ${\displaystyle X}$ is given, then the cover admits some Lebesgue's number ${\displaystyle \delta >0}$.

The notion of Lebesgue's numbers itself is useful in other applications as well.