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Difference between revisions of "Baire space"

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{{MSC|54A05}}
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[[Category:Topology]]
 
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Any space in which the [[Baire theorem|Baire theorem]] on complete spaces is valid.
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Any space in which the [[Baire theorem|Baire category theorem]] on complete metric spaces is valid.
  
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The metric space the points of which are infinite sequences $\{n_i\}=\{n_1,n_2,\dotsc\}$ of natural numbers, and the distance is given by the formula:
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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:
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\[
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\rho(\{n_i\},\{m_i\}) = \frac1{k_0}\, .
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\]
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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,
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totally disconnected and with no isolated points.
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Observe that the Baire space is the topological product of countably many
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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.
  
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$$\rho(\{n_i\},\{m_i\}) = \frac1{k_0}$$
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====Comments====
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By the [[Baire theorem|Baire category theorem]] the latter space is a Baire space in the sense of the first definition.
  
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where $k_0$ is the first natural number $k$ for which $n_k\neq m_k$. This is a complete metric separable zero-dimensional space containing the topological image of any metric separable zero-dimensional space.
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====References====
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{|
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|valign="top"|{{Ref|Kec}}|| A. S. Kechris, "Classical Descriptive Set Theory", Springer (1994)
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|-
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|valign="top"|{{Ref|Kel}}||  J.L. Kelley,  "General topology" , v. Nostrand  (1955)
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|-
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|valign="top"|{{Ref|Ox}}|| J.C. Oxtoby,  "Measure and category" , Springer  (1971)
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|}

Revision as of 07:39, 1 August 2012

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

Any space in which the Baire category theorem on complete metric spaces is valid.

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)
How to Cite This Entry:
Baire space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Baire_space&oldid=20899
This article was adapted from an original article by P.S. Aleksandrov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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