Analysis MOC

Lebesgue number

Let be a compact metric space and be an open cover. Then there exists a Lebesgue number such that every with diameter less than is contained entirely within one of the covering sets, anal i.e.

Proof

Since is an open cover, for every there exists some neighbourhood of , and hence some so that . Then is an open cover, and since is compact there exists some finite subcover where are points in . Then is a Lebesgue number.

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