1. The Baire category theorem - Aarhus Universitet

On locally compact Hausdorff spaces
Proof.
Since X is locally compact we can, for each k ∈ K, find an open
set Wk such that k ∈ Wk and Wk is compact.
Since {Wk : k ∈ K} is an open cover of K, and K is compact,
there is a finite collection k1,k2,...,kN such that
K ⊆ Wk1 ∪ Wk2 ∪ · · · ∪ WkN . Set W = Wk1 ∪ Wk2 ∪ · · · ∪ WkN and
note that W ⊆ Wk1 ∪ Wk2 ∪ · · · ∪ WkN .
It follows from Lemma 3 that W is compact. This completes the
proof in the case where U = X, since we can then take V = W.
When U = X, let C = Uc be the complement of U in X. For each
p ∈ C there is an open set W ′ p such that K ⊆ W ′ p and p /∈ W ′ p .
This follows from Lemma 1.
Note that
C ∩ W ∩ W ′ p = ∅,
p∈C
and that each set C ∩ W ∩ W ′ p is compact.
Klaus Thomsen 1. The Baire category theorem