Baire's theorem | category theorem

Baire's category theorem, also known as Baire's theorem and the category theorem, is a result in analysis and set theory which roughly states that in certain spaces, the intersection of any countable collection of "large" sets remains "large." The appearance of "category" in the name refers to the interplay of the theorem with the notions of sets of first and second category. Precisely stated, the theorem says that if a space S is either a complete metric space or a locally compact T^2-space, then the intersection of every countable collection of dense open subsets of S is necessarily dense in S.

complete metric space | countable set | first category | meager set | metric space | nonmeager set | nowhere dense | second category | T_2-space