Editing Baire category theorem (section)

==Uses==
In [[functional analysis]], '''BCT1''' can be used to prove the [[open mapping theorem (functional analysis)|open mapping theorem]], the [[closed graph theorem]] and the [[uniform boundedness principle]].

'''BCT1''' also shows that every nonempty complete metric space with no [[isolated point]] is [[uncountable]]. (If <math>X</math> is a nonempty countable metric space with no isolated point, then each [[singleton (mathematics)|singleton]] <math>\{x\}</math> in <math>X</math> is [[nowhere dense]], and <math>X</math> is [[meagre set|meagre]] in itself.) In particular, this proves that the set of all [[real number]]s is uncountable.

'''BCT1''' shows that each of the following is a Baire space:
* The space <math>\R</math> of [[real number]]s
* The [[irrational number]]s, with the metric defined by <math>d(x, y) = \tfrac{1}{n+1},</math> where <math>n</math> is the first index for which the [[continued fraction]] expansions of <math>x</math> and <math>y</math> differ (this is a complete metric space)
* The [[Cantor set]]

By '''BCT2''', every finite-dimensional Hausdorff [[manifold]] is a Baire space, since it is locally compact and Hausdorff. This is so even for non-[[paracompact]] (hence nonmetrizable) manifolds such as the [[long line (topology)|long line]].

'''BCT''' is used to prove [[Hartogs's theorem]], a fundamental result in the theory of several complex variables.

'''BCT1''' is used to prove that a [[Banach space]] cannot have countably infinite dimension.