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Every complete metric space is a Baire space.
Complete metric space is Baire: Every complete metric space is a Baire space; equivalently, in a complete metric space, every countable intersection of dense open sets is dense.
This is the standard formulation of the Baire category theorem in the language of Baire spaces, and it underlies many arguments involving meager sets and nowhere dense sets.
A complete metric space is a metric space in which every Cauchy sequence converges (as a convergent sequence) to a point of .
A Baire space is a topological space such that for every sequence of open sets that are dense, the intersection is dense in .
A dense set in a topological space is a subset such that its closure is all of , i.e. .
An open set in a topological space is a subset such that .
Open sets are the basic “observable” sets in topology: they define neighborhoods of points and determine operations like interior and closure. They also control continuity through preimages.
Baire category theorem: Let be a complete metric space. If is a sequence of dense open sets in , then the intersection is dense in .
A meager set in a topological space is a subset that can be written as a countable union of nowhere dense sets.
Meager sets are also called “sets of first category.” Their complements are the residual sets; in a Baire space, meager sets are topologically “small.”