For each open covering exists a number so that each subset with diameter in a coverage amount is included, ie . Such a number is called the Lebesgue number of the coverage for .
Every smaller number is of course also a Lebesgue number for this coverage and this space.
proof
If so, any number can be chosen, since all subsets are contained in a coverage set.
So be now . As is compact, can be made choose a finite sub-covering, so be a (finite) covering of X .
For all , set and define a function by .
For any, but fixed, choose now so that . Choose now a small enough so that the environment of lies in the selected coverage amount, ie . Well is so is . The function is therefore very positive.
Since continuous and is defined on a compact , it assumes a minimum . This is the Lebesgue number you are looking for:
Let a subset with diameter be smaller . For each is now in the neighborhood of . Now choose any one .
Now be chosen so that for becomes a maximum. Is now , and the environment of , and thus are very out of coverage . So now one with the property of the Lebesgue number is found.