Namespaces
Variants
Actions

Baire space

From Encyclopedia of Mathematics
Jump to: navigation, search
The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.

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
How to Cite This Entry:
Baire space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Baire_space&oldid=27641
This article was adapted from an original article by P.S. Aleksandrov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article