Let be a compactmetric 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, #m/thm/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.