1. The Baire category theorem - Aarhus Universitet

On locally compact Hausdorff spaces
Proof.
Consider an element k ∈ K. Since X is Hausdorff there are open
sets, Wk ⊆ X and Uk ⊆ X, such that k ∈ Wk,p ∈ Uk and
Wk ∩ Uk = ∅.
Note that the collection {Wk : k ∈ K} is an open cover of K.
Since K is compact there is a finite collection k1,k2,... ,kN such
that
K ⊆ Wk1 ∪ Wk2 ∪ · · · ∪ WkN .
Klaus Thomsen 1. The Baire category theorem