Baire space

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]).

Comments

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

References

[Mo]	Y. Moschovakis, "Descriptive set theory". Studies in Logic and the Foundations of Mathematics, 100. North-Holland Publishing Co., Amsterdam-New York, 1980. MR0561709 Zbl 0433.03025
[Bo]	N. Bourbaki, "General topology: Chapters 5-10", Elements of Mathematics (Berlin). Springer-Verlag, Berlin, 1998. MR1726872 Zbl 0894.54002
[Ke]	J.L. Kelley, "General topology" , v. Nostrand (1955) MR0070144 Zbl 0066.1660
[Ox]	J.C. Oxtoby, "Measure and category" , Springer (1971) MR0393403 0217.09201 Zbl 0217.09201