Baire Category Theorem

A theorem in topology and functional analysis stating that in a complete metric space (or more generally a Baire space) the countable intersection of dense open sets is dense; equivalently, the space cannot be written as a countable union of nowhere-dense sets.

The theorem states that complete metric spaces and locally compact Hausdorff spaces are Baire spaces: the countable intersection of dense open sets is dense (equivalently, nonempty open sets are not meager).