Baire category and complete spaces

A set is nowhere dense when its closure has empty interior. This is stronger than merely having empty interior: the rationals have empty interior in the real line, but their closure is the whole line, so they are not nowhere dense. A singleton in the real line is nowhere dense because it is closed and contains no interval. A set is meager if it is a countable union of nowhere dense sets. Thus the rationals are meager, even though they are dense. These examples show that density and category-smallness answer different questions.

The complement of a meager set is called residual or comeager. A residual set can be described as containing a countable intersection of dense open sets: replace each nowhere dense set by its closure and take complements. Some authors use generic to mean residual, so the intended definition should be stated. Category is not cardinality or measure. For example, a countable dense set can be topologically small, while a closed nowhere dense set can still have positive Lebesgue measure. No probability law is part of the definition of category.

Let X be a complete metric space and U_n a sequence of open dense subsets. To show their intersection is dense, start inside any nonempty open region W. Choose a nonempty closed ball of positive radius contained in W intersected with U_1, with radius at most one half. Inside its open interior choose a smaller closed ball contained also in U_2, with radius at most one quarter. Continue, nesting the closed balls and choosing radii tending to zero. Openness gives room for each ball, while density ensures there is a point at which to place it.

The centers form a Cauchy sequence because all late centers lie in any fixed earlier ball and those ball diameters tend to zero. Completeness supplies a limit in X. Each earlier ball is closed, so that limit lies in every ball and hence in W and every U_n. This proves density of the intersection. The limit argument uses neither compactness of the balls nor local compactness of X. In a general complete metric space, closed bounded balls need not be compact, but their shrinking nested structure is sufficient here.

Equivalently, a nonempty complete metric space cannot be a countable union of nowhere dense subsets, and no nonempty open subset is meager in that space. The ambient qualification matters. The rational numbers with usual distance are a countable union of their singletons, each nowhere dense relative to the rationals, and the whole rational space is nonempty open in itself. It is therefore not a Baire space and cannot be complete. This is an application of the theorem's contrapositive, not a claim that every incomplete metric space must fail the Baire property.

On the real line, enumerate the rationals q_n. Every complement of a singleton {q_n} is open dense. Their intersection is the irrational numbers, so Baire gives another proof that the irrationals are dense. The theorem ensures a point survives countably many closed nowhere dense exclusions even when a drawing shows only finitely many. It does not directly handle an uncountable intersection: intersecting the complements of all real singletons gives the empty set. Countability is a working part of the construction, because one obstruction is addressed at each step of a sequence.