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Difference between revisions of "Baire space"

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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 theorem|Baire category theorem]], any [[Complete metric space|complete metric space]] is a Baire space. Another class of Baire spaces are [[Locally compact space|locally compact]] [[Hausdorff space|Hausdorff spaces]].
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Any space in which the intersection of any countable family of dense open subsets is dense (cp. with Section 9 of {{Cite|Ox}}
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and Definition 3 in Section 5.3 of Chapter IX in {{Cite|Bo}}). An open set of a Baire space is itself a Baire space. By the [[Baire theorem|Baire category theorem]], any [[Complete metric space|complete metric space]] is a Baire space. Another class of Baire spaces are [[Locally compact space|locally compact]] [[Hausdorff space|Hausdorff spaces]] (see Section 5.3 of Chapter IX in {{Cite|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:
 
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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\rho(\{n_i\},\{m_i\}) = \frac1{k_0}\, .
 
\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 space|separable]] and [[Dimension|zero-dimensional]], [[Totally-disconnected space|totally disconnected]] and with no isolated points. Observe that the Baire space is the [[Topological product|topological product]] of countably many copies  of the natural numbers $\mathbb N$ endowed with the [[Discrete topology|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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where $k_0$ is the first natural number $k$ for which $n_k\neq m_k$ (see for instance Section 1A of {{Cite|Mo}}). Such metric is complete and the space is [[Separable space|separable]] and [[Dimension|zero-dimensional]], [[Totally-disconnected space|totally disconnected]] and with no isolated points. Observe that the Baire space is the [[Topological product|topological product]] of countably many copies  of the natural numbers $\mathbb N$ endowed with the [[Discrete topology|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 {{Cite|Mo}}).
  
 
====Comments====
 
====Comments====
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====References====
 
====References====
 
{|
 
{|
|valign="top"|{{Ref|Kec}}|| A. S. Kechris, "Classical Descriptive Set Theory", Springer (1994)
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|valign="top"|{{Ref|Mo}}|| Y. Moschovakis, "Descriptive set theory". Studies in Logic and the Foundations of Mathematics, 100. North-Holland Publishing Co., Amsterdam-New York,  1980. {{MR|0561709}} {{ZBL|0433.03025}}
 
|-
 
|-
|valign="top"|{{Ref|Kel}}|| J.L. Kelley,   "General topology" , v. Nostrand (1955)
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|valign="top"|{{Ref|Bo}}|| N. Bourbaki, "General topology: Chapters  5-10", Elements of Mathematics (Berlin). Springer-Verlag, Berlin,  1998{{MR|1726872}} {{ZBL|0894.54002}}
 
|-
 
|-
|valign="top"|{{Ref|Ox}}|| J.C. Oxtoby,  "Measure and category" , Springer  (1971)
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|valign="top"|{{Ref|Ke}}||  J.L. Kelley,  "General topology" , v. Nostrand  (1955) {{MR|0070144}} {{ZBL|0066.1660}}
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|-
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|valign="top"|{{Ref|Ox}}|| J.C. Oxtoby,  "Measure and category" , Springer  (1971) {{MR|0393403}} {{ZBL| 0217.09201}}
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|-
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|}
 
|}

Revision as of 08:11, 17 August 2012

2020 Mathematics Subject Classification: Primary: 54A05 [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=27610
This article was adapted from an original article by P.S. Aleksandrov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article