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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 $\mathbb{R}$ 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 $\mathcal{T}$ 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 $\mathcal{T}$ 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 1 topological space $\mathcal{T}$ .
Footnotes
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- Some authors only define a locally compact space to be a Hausdorff space; that is the sense required for this theorem.
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