- #1
Useful nucleus
- 370
- 58
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.
(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.