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
.
The key observation is that
.
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
Munkres, James R. (1974), Topology: A first course, p. 179, ISBN 978-0-13-925495-6