Baire category theorem

In a non-empty complete metric space, any countable intersection of dense, open subsets is non-empty.

In fact, such countable intersections of dense, open subsets are dense. So the theorem holds also for any non-empty open subset of a complete metric space.

Alternative formulations:Call a set first category, or a meagre set, if it is a countable union of nowhere dense sets, otherwise second category. The Baire category theorem is often stated as “no non-empty complete metric space is of first category”, or, trivially, as “a non-empty, complete metric space is of second category”. In short, this theorem says that every nonempty complete metric space is a Baire space.

In functional analysis, this important property of complete metric spaces forms the basis for the proofs of the important principles of Banach spaces: the open mapping theorem and the closed graph theorem.

It may also be taken as giving a concept of “small sets”, similar to sets of measure zero: a countable union of these sets remains “small”. However, the real line $ℝ$ may be partitioned into a set of measure zero and a set of first category; the two concepts are distinct.

Note that, apart from the requirement that the set be a complete metric space, all conditions and conclusions of the theorem are phrased topologically. This “metric requirement” is thus something of a disappointment. As it turns out, there are two ways to reduce this requirement.

First, if a topological space $𝒯$ is homeomorphic to a non-empty open subset of a complete metric space, then we can transfer the Baire property through the homeomorphism, so in $𝒯$ too any countable intersection of open dense sets is non-empty (and, in fact, dense). The other formulations also hold in this case.

Second, the Baire category theorem holds for a locally compact, Hausdorff topological space $𝒯$.