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Good night, someone could help me with some interesting applications of this important theorem and also I could say how I draw a dense subset in the complete space used in the theorem. Thanks for your help

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marked as duplicate by Andrés E. Caicedo, Pete L. Clark, user61527, mrf, Shobhit Dec 7 '13 at 7:36

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The existence of a nowhere differentiable function. If you want to read it take a look at Munkres in chapter 49. It is a really fun excercise to show that $\mathbb{Q}$ is not a Baire space, whereas $\mathbb{R\backslash \mathbb{Q}}$ is.

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I don't know if this is the type of result that you're looking for, but you use Baire category to show that a Banach space cannot have a countably-infinite Hamel basis {$e_i$}: the span of a given basis element is nowhere-dense in the Banach space, so that the union of the span of the countably-many $e_i$ is a countable union of nowhere-dense sets, so that it cannot span a complete metric (normed) space--a Banach space.

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