Transcription of The Baire category theorem - UCL
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The Baire category theorem - UCL
MATHEMATICS 3103 (Functional Analysis)YEAR 2012 2013, TERM 2 HANDOUT #7: THE Baire category theorem AND ITSCONSEQUENCESWe shall begin this last section of the course by returning tothe study of general metricspaces, and proving a fairly deep result called theBaire category shall thenapply the Baire category theorem to prove three fundamentalresults in functional analysis:the Uniform Boundedness theorem , the Open Mapping theorem ,and the Closed Baire category theoremLetXbe a metric space . A subsetA Xis callednowhere denseinXif the interior ofthe closure ofAis empty, (A) = . Otherwise put,Ais nowhere dense iff it is containedin a closed set with empty interior. Passing to complements,we can say equivalently thatAis nowhere dense iff its complement contains a dense open set (why?).Proposition a metric space . Then:(a) Any subset of a nowhere dense set is nowhere dense.(b) The union of finitely many nowhere dense sets is nowhere dense.(c) The closure of a nowhere dense set is nowhere dense.
We are now ready to state the Baire category theorem: Theorem 7.3 (Baire category theorem) Let X be a complete metric space. Then: (a) A meager set has empty interior. (b) The complement of a meager set is dense. (That is, a residual set is dense.) (c) A countable intersection of dense open sets is dense.
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