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
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: The notion of Lebesgue's numbers itself is useful in other applications as well.
be an open cover of
is compact we can extract a finite subcover
will serve as a Lebesgue's number.
is not empty, and define a function
is continuous on a compact set, it attains a minimum
δ
The key observation is that, since every
, the extreme value theorem shows
is the desired Lebesgue's number.
, then by definition of diameter,
denotes the ball of radius
Suppose for contradiction that
is sequentially compact,
is an open cover of
, and the Lebesgue number
This enables us to perform the following construction:
It is therefore possible by the axiom of choice to construct a sequence
is sequentially compact, there exists a subsequence
is an open cover, there exists some
is open, there exists
Now we invoke the convergence of the subsequence
Finally, define
by the triangle inequality, which implies that
This yields the desired contradiction.
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