# Baire space

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2020 Mathematics Subject Classification: Primary: 54E52 [MSN][ZBL]

Any space in which the intersection of any countable family of dense open subsets is dense (cp. with Section 9 of [Ox] and Definition 3 in Section 5.3 of Chapter IX in [Bo]). An open set of a Baire space is itself a Baire space. By the Baire category theorem, any complete metric space is a Baire space. Another class of Baire spaces are locally compact Hausdorff spaces (see Section 5.3 of Chapter IX in [Bo]).

The name is also used for the metric space consisting of infinite sequences $\{n_i\}=\{n_1,n_2,\dotsc\}$ of natural numbers, with the distance given by the following formula: $\rho(\{n_i\},\{m_i\}) = \frac1{k_0}\, .$ where $k_0$ is the first natural number $k$ for which $n_k\neq m_k$ (see for instance Section 1A of [Mo]). Such metric is complete and the space is separable and zero-dimensional, totally disconnected and with no isolated points. Observe that the Baire space is the topological product of countably many copies of the natural numbers $\mathbb N$ endowed with the discrete topology. Moreover it is homeomorphic to the irrational numbers endowed with the topology of subset of $\mathbb R$. Any zero-dimensional separable metric space of dimension zero can be embedded in the Baire space. Moreover, for every Polish space $\mathcal{M}$ there is a continuous surjection from the Baire space onto $\mathcal{M}$ (see Theorem 1A.1 of [Mo]).