# Computing lebesgue number for an open covering

1. Mar 26, 2013

### Useful nucleus

Munkres proof for the Lebesgue number lemma which is
(If X is a compact metric space and A is an open covering then there exists δ>0 such that for each subset of X having diameter less than δ , then there exists an element of the covering A containing it)
gives a way to compute δ using a finite subcollection that covers the compact metric space. However, this is not necessarily the smallest Lebesgue number. I wonder if there is another proof that involves evaluating the smallest Lebesgue number, if the latter exists.

2. Mar 29, 2013

### lugita15

Useful nucleus, do you mean the largest Lebesgue number? Any positive number smaller than a Lebesgue number for a covering is also a Lebesgue number for the covering.

3. Apr 1, 2013

### Useful nucleus

lugita15, you are absolutely right, I meant largest Lebesgue number if exist. I did not have time to refelct on this question again, but will give it a try soon.