Lebesgue's number lemma

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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 is compact and an open cover of is given, then there exists a number such that every subset of having diameter less than is contained in some member of the cover.

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.


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 not 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 (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, .


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