This theorem's namesake gives his name to almost-open sets.
In 1977 Charles Blair demonstrated that for complete metric spaces, this theorem is equivalent to the axiom of dependent choice. This theorem is used to prove that pointwise boundedness of a family of continuous linear operators from a Banach space to a normed space entail uniform boundedness; it is also used to prove that a continuous linear map between two Banach spaces is an open map. A way to prove this theorem is by constructing a nested sequence of closed spheres with radii converging to zero, which contains a single point, contradicting the terms of the construction. This theorem's namesake gives his name to almost-open sets. A corollary of this theorem is that a complete metric space without isolated points is uncountable. A complete metric space cannot be represented as a countable union of nowhere dense sets according to, for 10 points, what man's "category" theorem?
In 1977 Charles Blair demonstrated that for complete metric spaces, this theorem is equivalent to the axiom of dependent choice.
Baire category theorem [prompt on category theorem]