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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<strong>The</strong> <strong>Baire</strong>’s <strong>category</strong> <strong>theorem</strong><br />
<strong>The</strong> <strong>Baire</strong>’s <strong>category</strong> <strong>theorem</strong>.<br />
If S is either<br />
(a) a complete metric space, or<br />
(b) a locally compact Hausdorff space,<br />
then the intersection of every countable collection of dense open<br />
sets in S is dense is S.<br />
Thus if Ei,i = 1,2,3,... , are dense and open in S, then ∞<br />
i=1 Ei is<br />
dense in S.<br />
Corollary.<br />
A complete metric space or a locally compact Hausdorff space is of<br />
the second <strong>category</strong>; thus they are not the union of a countable<br />
collection of nowhere dense subsets.<br />
Klaus Thomsen <strong>1.</strong> <strong>The</strong> <strong>Baire</strong> <strong>category</strong> <strong>theorem</strong>