Proof That Disjoint Compact Subsets of a Hausdorff Space are Contained in Disjoint Open Subsets
Theorem
Let
and
be disjoint compact subsets of a Hausdorff space
Open sets
and
exist such that
and
Proof
Let
Since
is compact so open sets
and
exist such that
and
(1)
The family of open sets
forms an open cover of the compact setso a finite subcover
exists. For each
we can find a correspondingsatisfying (1).
Let
and
Then
and
and bothandare open.
For each
hence