Theorem
Letand
be disjoint compact subsets of a Hausdorff space
Open sets
and
exist such that
and
Proof
LetSince
is compact so open sets
and
exist such that
and
(1)
The family of open setsforms an open cover of the compact set
so a finite subcover
exists. For each
we can find a corresponding
satisfying (1).
Letand
Thenand
and both
and
are open.
For eachhence