Explore: Baire Spaces

According to the Baire category theorem, compact Hausdorff spaces and complete metric spaces are examples of Baire spaces. The Baire category theorem

which gives sufficient conditions for a topological space to be a Baire space (a topological space such that the intersection of countably many dense open

theory, the Baire space is the set of all infinite sequences of natural numbers with a certain topology, called the product topology. This space is commonly

René-Louis Baire (French: [bɛʁ]; 21 January 1874 – 5 July 1932) was a French mathematician most famous for his Baire category theorem, which helped to

complete (pseudo)metric spaces as well as Hausdorff locally compact spaces are Baire spaces, they are also nonmeagre spaces. Any subset of a meagre set

group that is also a Baire space, then G is locally compact. This shows that for Hausdorff topological groups that are also Baire spaces, σ-compactness implies

can also define the arithmetic hierarchy of subsets of the Cantor and Baire spaces relative to some set of natural numbers. In fact boldface Σ n 0 {\displaystyle

one of these spaces. A similar extension is possible for countable powers and to products of powers of Cantor space and powers of Baire space. As is the

In mathematics, computable numbers are the real numbers that can be computed to within any desired precision by a finite, terminating algorithm. They are

those spaces that can be written as an intersection of countably many open subsets of some complete metric space. Since the conclusion of the Baire category