The Baire Category Theorem is a fundamental result in functional analysis and topology, particularly in the study of complete metric spaces and topological spaces. It provides insight into the structure of certain types of sets and establishes the notion of "largeness" in the context of topological spaces. The theorem states that in a complete metric space (or, more generally, a Baire space), the intersection of countably many dense open sets is dense.
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Every complete metric space is a Baire space: every countable intersection of open dense subsets is dense. Equivalently, no nonempty open subset is a countable union of nowhere-dense subsets.
To prove the dense-intersection form, let be nonempty and open and let be open dense sets. Successively choose closed ballsstarting with a ball contained in . The centres form a Cauchy sequence. Its limit belongs to every closed ball, and hence to .