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''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024090/c0240901.png" />''
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A separated [[Complete uniform space|complete uniform space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024090/c0240902.png" /> for which there exists a uniformly-continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024090/c0240903.png" /> such that for any uniformly-continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024090/c0240904.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024090/c0240905.png" /> into a separated complete uniform space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024090/c0240906.png" /> there exists a unique uniformly-continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024090/c0240907.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024090/c0240908.png" />. The subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024090/c0240909.png" /> is dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024090/c02409010.png" /> and the image of entourages in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024090/c02409011.png" /> under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024090/c02409012.png" /> are entourages in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024090/c02409013.png" />; their closures in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024090/c02409014.png" /> constitute a fundamental system of entourages in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024090/c02409015.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024090/c02409016.png" /> is separated, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024090/c02409017.png" /> is injective (this allows one to identify <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024090/c02409018.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024090/c02409019.png" />). The separated completion of a subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024090/c02409020.png" /> is isomorphic to the closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024090/c02409021.png" />. The separated completion of a product of uniform spaces is isomorphic to the product of the separated completions of the spaces occurring as factors in the product.
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The proof of the existence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024090/c02409022.png" /> generalizes in fact Cantor's construction of the set of real numbers from the set of rational numbers.
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'' $  X $''
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A separated [[Complete uniform space|complete uniform space]]  $  \widehat{X}  $
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for which there exists a uniformly-continuous mapping  $  i :  X \rightarrow \widehat{X}  $
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such that for any uniformly-continuous mapping  $  f $
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from  $  X $
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into a separated complete uniform space  $  Y $
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there exists a unique uniformly-continuous mapping  $  g :  \widehat{X}  \rightarrow Y $
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with  $  f = g \circ i $.
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The subspace  $  i ( X) $
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is dense in  $  \widehat{X}  $
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and the image of entourages in  $  X $
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under  $  i \times i $
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are entourages in  $  i ( X) $;
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their closures in  $  \widehat{X}  \times \widehat{X}  $
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constitute a fundamental system of entourages in  $  \widehat{X}  $.
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If  $  X $
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is separated, then  $  i $
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is injective (this allows one to identify  $  X $
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with  $  i ( X) $).
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The separated completion of a subspace  $  A \subset  X $
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is isomorphic to the closure of  $  i ( A) \subset  \widehat{X}  $.
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The separated completion of a product of uniform spaces is isomorphic to the product of the separated completions of the spaces occurring as factors in the product.
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The proof of the existence of $  \widehat{X}  $
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generalizes in fact Cantor's construction of the set of real numbers from the set of rational numbers.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. General topology" , Addison-Wesley  (1966)  (Translated from French)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. General topology" , Addison-Wesley  (1966)  (Translated from French)</TD></TR></table>

Latest revision as of 17:46, 4 June 2020


$ X $

A separated complete uniform space $ \widehat{X} $ for which there exists a uniformly-continuous mapping $ i : X \rightarrow \widehat{X} $ such that for any uniformly-continuous mapping $ f $ from $ X $ into a separated complete uniform space $ Y $ there exists a unique uniformly-continuous mapping $ g : \widehat{X} \rightarrow Y $ with $ f = g \circ i $. The subspace $ i ( X) $ is dense in $ \widehat{X} $ and the image of entourages in $ X $ under $ i \times i $ are entourages in $ i ( X) $; their closures in $ \widehat{X} \times \widehat{X} $ constitute a fundamental system of entourages in $ \widehat{X} $. If $ X $ is separated, then $ i $ is injective (this allows one to identify $ X $ with $ i ( X) $). The separated completion of a subspace $ A \subset X $ is isomorphic to the closure of $ i ( A) \subset \widehat{X} $. The separated completion of a product of uniform spaces is isomorphic to the product of the separated completions of the spaces occurring as factors in the product.

The proof of the existence of $ \widehat{X} $ generalizes in fact Cantor's construction of the set of real numbers from the set of rational numbers.

References

[1] N. Bourbaki, "Elements of mathematics. General topology" , Addison-Wesley (1966) (Translated from French)
How to Cite This Entry:
Completion of a uniform space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Completion_of_a_uniform_space&oldid=46426
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article