Namespaces
Variants
Actions

Difference between revisions of "Baire space"

From Encyclopedia of Mathematics
Jump to: navigation, search
m (Addred links.)
Line 3: Line 3:
 
{{TEX|done}}
 
{{TEX|done}}
  
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
+
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]].
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:
 
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:
Line 10: Line 9:
 
\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 and zero-dimensional,
+
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.
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====
 
====Comments====

Revision as of 15:43, 8 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. 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)
How to Cite This Entry:
Baire space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Baire_space&oldid=27436
This article was adapted from an original article by P.S. Aleksandrov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article