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 U be an open cover of X . Since X is compact we can extract a finite subcover { A 1 , … , A n } ⊆ U .
For each i ∈ { 1 , … , n } , let C i := X ∖ A i and define a function f : X → R by f ( x ) := 1 n ∑ i = 1 n d ( x , C i ) .
Since f is continuous on a compact set, it attains a minimum δ . The key observation is that δ > 0 . If Y is a subset of X of diameter less than δ , then there exist x 0 ∈ X such that Y ⊆ B δ ( x 0 ) , where B δ ( x 0 ) denotes the ball of radius δ centered at x 0 (namely, one can choose as x 0 any point in Y ). Since f ( x 0 ) ≥ δ there must exist at least one i such that d ( x 0 , C i ) ≥ δ . But this means that B δ ( x 0 ) ⊆ A i and so, in particular, Y ⊆ A i .
