Baire space

2020 Mathematics Subject Classification: Primary: 54A05 [MSN][ZBL]

Any space in which the intersection of any countable family of dense open subsets is dense. 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.

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 $n_k\neq m_k$. 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.

Comments

By the Baire category theorem the latter space is a Baire space in the sense of the first definition.

References

[Kec]	A. S. Kechris, "Classical Descriptive Set Theory", Springer (1994)
[Kel]	J.L. Kelley, "General topology" , v. Nostrand (1955)
[Ox]	J.C. Oxtoby, "Measure and category" , Springer (1971)