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Analytic proof of the Baire category theorem

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Ways to Say you're done	The Baire category theorem and cardinality	Game proof of the Baire Category Theorem	first category
Baire category theorem	How much for the little girl?	nowhere dense	Cauchy sequence
GpBCT: proof that Bob wins on the complement of an open dense set	GpBCT: proof that Alice wins on an open set	second category	Open ball
Mathematical Induction	Banach space	GpBCT: proof that Bob wins on a countable union of sets if he's guaranteed a win on each one of them	empty set
dense	countable	Induction	Convergence
ball	subset

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