Baire space

In mathematics, a Baire space is a topological space such that every intersection of a countable collection of open dense sets in the space is also dense. Complete metric spaces and locally compact Hausdorff spaces are examples of Baire spaces according to the Baire category theorem. The spaces are named in honor of René-Louis Baire who introduced the concept. In mathematics, a Baire space is a topological space such that every intersection of a countable collection of open dense sets in the space is also dense. Complete metric spaces and locally compact Hausdorff spaces are examples of Baire spaces according to the Baire category theorem. The spaces are named in honor of René-Louis Baire who introduced the concept. In an arbitrary topological space, the class of closed sets with empty interior consists precisely of the boundaries of dense open sets. These sets are, in a certain sense, 'negligible'. Some examples are finite sets in ℝ, smooth curves in the plane, and proper affine subspaces in a Euclidean space. If a topological space is a Baire space then it is 'large', meaning that it is not a countable union of negligible subsets. For example, the three-dimensional Euclidean space is not a countable union of its affine planes.