§ Lebesgue number lemma (TODO)
for a compact space and an open cover , there is a radius
such that any ball of such a radius will be in some open cover: For all
, for all such balls , there exists a such that
. Intuitively, pick a point . for
each open , we have a ball that sits inside it since is open.
Find the largest such radius, we can do so since is the closed subset of a compact set.
This gives us a function that maps a point to the largest radius of ball
that can fit in some open cover around it. This function is a continuous function (why?)
on a compact set, and thus has a minimum. So, for all points , if you give a
ball of radius , I can find some open cover around it.
§ Lebesgue number lemma, Version 2:
for a compact space and an open cover , there is a diameter such that
set of smaller radius will be in some open cover: For all , for all
opens such that , there exists a such that .
If we can find radius that satisfies this, then if we are given a set of diameter less than ,
there will be a ball that contains the set of diameter at most ,
and this ball will be contained in some . So we will have the
§ Lebesgue number lemma, proof from Hatcher