Baire Category Theorem - Uses of The Theorem

Uses of The Theorem

BCT1 is used in functional analysis to prove the open mapping theorem, the closed graph theorem and the uniform boundedness principle.

BCT1 also shows that every complete metric space with no isolated points is uncountable. (If X is a countable complete metric space with no isolated points, then each singleton {x} in X is nowhere dense, and so X is of first category in itself.) In particular, this proves that the set of all real numbers is uncountable.

BCT1 shows that each of the following is a Baire space:

• The space R of real numbers
• The irrational numbers, with the metric defined by d(x, y) = 1 / (n + 1), where n is the first index for which the continued fraction expansions of x and y 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.