1. The Baire category theorem - Aarhus Universitet
1. The Baire category theorem - Aarhus Universitet
1. The Baire category theorem - Aarhus Universitet
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On locally compact Hausdorff spaces<br />
Proof.<br />
Since X is locally compact we can, for each k ∈ K, find an open<br />
set Wk such that k ∈ Wk and Wk is compact.<br />
Since {Wk : k ∈ K} is an open cover of K, and K is compact,<br />
there is a finite collection k1,k2,...,kN such that<br />
K ⊆ Wk1 ∪ Wk2 ∪ · · · ∪ WkN . Set W = Wk1 ∪ Wk2 ∪ · · · ∪ WkN and<br />
note that W ⊆ Wk1 ∪ Wk2 ∪ · · · ∪ WkN .<br />
It follows from Lemma 3 that W is compact. This completes the<br />
proof in the case where U = X, since we can then take V = W.<br />
Klaus Thomsen <strong>1.</strong> <strong>The</strong> <strong>Baire</strong> <strong>category</strong> <strong>theorem</strong>