Baire category theorem

A complete metric space cannot be covered by countably many nowhere-dense sets. Equivalently, every countable intersection of dense open sets is dense.

Nowhere-dense set: A set whose closure has empty interior. The theorem says countably many such sets cannot cover a complete metric space.
Metric space: A set equipped with a distance function satisfying the metric axioms. The theorem’s standard form concerns complete metric spaces.
Baire space: A topological space in which countable intersections of dense open sets are dense. Complete metric spaces are Baire spaces, making the theorem useful beyond a single metric presentation.
René-Louis Baire: A French mathematician whose work on discontinuous functions and category gave rise to the Baire category theorem. The theorem is named for Baire, who established its foundational form.
First category set: A set that is a countable union of nowhere-dense subsets of a topological space. The theorem says a complete metric space itself cannot be first category in itself.
Dense open set: An open subset whose closure is the entire space. The theorem’s equivalent form says countable intersections of dense open sets remain dense.
Closure (topology): The smallest closed set containing a given subset of a topological space. Nowhere-density is defined by requiring the closure to have empty interior.
Uniform boundedness principle: A theorem stating that pointwise-bounded families of continuous linear operators are uniformly bounded under suitable completeness assumptions. Its proof uses the Baire category theorem to obtain a neighborhood with a uniform bound.
Baire's thesis: René-Louis Baire’s 1899 doctoral thesis on discontinuous functions and the classification of functions. The theorem emerged from the thesis’s study of limits and exceptional sets of functions.
Second category set: A set that is not a countable union of nowhere-dense subsets in the ambient space. Every nonempty complete metric space is second category in itself.

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