1. The Baire category theorem - Aarhus Universitet

On locally compact Hausdorff spaces
Proof.
It follows from Lemma 2 that there is a finite set, p1,p2,... ,pN,
such that
N
C ∩ W ∩ W ′ pi = ∅. (2)
i=1
Set V = W ∩ W ′ p1 ∩ W ′ p2 ∩ · · · ∩ W ′ pN . Then
K ⊆ V ⊆ V ⊆ W ∩ W ′ p1 ∩ W ′ p2 ∩ · · · ∩ W ′ pN . It follows from (2)
that W ∩ W ′ p1 ∩ W ′ p2 ∩ · · · ∩ W ′ pN ⊆ U and from Lemma 3 that V
is compact.
Klaus Thomsen 1. The Baire category theorem