§ Compact Hausdorff spaces are normal
Let be two disjoint closed subsets. We wish to exhibit disjoint opens
which separate . Formally, we want .
The crucial idea is to take all pairs of points in , and use
Hausdorffness to find opens
such that . which
separate all pairs and , and then to use compactness to escalate this
into a real separating cover.
Now that we have the pairs, for a fixed , consider the cover
. This covers the set , hence there is a finite subcover
. Now, we go back, and build the set
. This is the intersection of a finite
number of opens, and is hence open. So we now have two sets and
which separate from . We can build such a pair that separates
each from all of . Then, using compactness again, we find a finite subcover of
sets such that the cover , each of the
cover (so covers ). This gives
us our final opens and . that separate and .