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 />
Before we continue the proof of <strong>Baire</strong>’s <strong>category</strong> <strong>theorem</strong> we<br />
develop a little of the general theory of locally compact Hausdorff<br />
spaces.<br />
Recall that a topological space is locally compact when every point<br />
has an open neighborhood with compact closure.<br />
Lemma<br />
Let X be a Hausdorff space, K a compact subset of X and p a<br />
point in the complement K c = X \K of K. <strong>The</strong>re are then open<br />
sets U and W such that p ∈ U,K ⊆ W and U ∩ W = ∅.<br />
Klaus Thomsen <strong>1.</strong> <strong>The</strong> <strong>Baire</strong> <strong>category</strong> <strong>theorem</strong>