Difference between revisions of "Baire space"
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Any space in which the [[Baire theorem|Baire theorem]] on complete spaces is valid. | Any space in which the [[Baire theorem|Baire theorem]] on complete spaces is valid. | ||
| − | The metric space the points of which are infinite sequences | + | 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: |
| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"> | + | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;">$\rho(\{n_i\},\{m_i\}) = \frac1{k_0}$</td> </tr></table> |
| − | where | + | 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. |
Revision as of 12:13, 8 February 2012
Any space in which the Baire theorem on complete spaces is valid.
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:
| $\rho(\{n_i\},\{m_i\}) = \frac1{k_0}$ |
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.
How to Cite This Entry:
Baire space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Baire_space&oldid=20898
Baire space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Baire_space&oldid=20898
This article was adapted from an original article by P.S. Aleksandrov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article