Charles Blair demonstrated that this theorem implies the axiom of dependent choice, a weaker version of the axiom of choice. A typical proof of this theorem constructs a nested sequence of open balls disjoint from a sequence of nowhere dense sets and uses the nested interval theorem to show the ball
Charles Blair demonstrated that this theorem implies the axiom of dependent choice, a weaker version of the axiom of choice. A typical proof of this theorem constructs a nested sequence of open balls disjoint from a sequence of nowhere dense sets and uses the nested interval theorem to show the balls have a non-empty intersection. This theorem states that the countable intersection of open, (*) dense subsets of a complete metric space, X, is dense, from which it follows that X is not a countable union of nowhere dense sets. This theorem is of fundamental importance because it can be used to prove the closed graph theorem, uniform boundedness principle, and open mapping theorem. For 10 points, name this "category theorem" fundamental to functional analysis.
Charles Blair demonstrated that this theorem implies the axiom of dependent choice, a weaker version of the axiom of choice.
Baire's category theorem [or: BCT]