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Difference between revisions of "Absolute retract for normal spaces"

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A topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010340/a0103401.png" /> such that every mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010340/a0103402.png" /> of any closed subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010340/a0103403.png" /> of an arbitrary [[Normal space|normal space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010340/a0103404.png" /> can be extended to the entire space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010340/a0103405.png" />. A direct product of absolute retracts for normal spaces is an absolute retract for normal spaces, as is any retract (cf. [[Retract of a topological space|Retract of a topological space]]) of an absolute retract for normal spaces. In particular, the following spaces are absolute retracts for normal spaces: the unit interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010340/a0103406.png" />; the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010340/a0103407.png" />-dimensional cube <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010340/a0103408.png" />; and the Hilbert cube <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010340/a0103409.png" />. Any two mappings of a binormal space into an absolute retract for normal spaces are homotopic; while a binormal absolute retract for normal spaces is contractible into a point.
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A topological space $X$ such that every mapping $g\colon A\to X$ of any closed subset $A$ of an arbitrary [[Normal space|normal space]] $Y$ can be extended to the entire space $Y$. A direct product of absolute retracts for normal spaces is an absolute retract for normal spaces, as is any retract (cf. [[Retract of a topological space|Retract of a topological space]]) of an absolute retract for normal spaces. In particular, the following spaces are absolute retracts for normal spaces: the unit interval $I$; the $n$-dimensional cube $I^n$; and the Hilbert cube $I^\omega$. Any two mappings of a binormal space into an absolute retract for normal spaces are homotopic; while a binormal absolute retract for normal spaces is contractible into a point.
  
  
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Usually an absolute retract (AR) for normal spaces is defined to be a normal space which is a retract of every normal space in which it is imbedded as a closed subset. A space satisfying the property in the article above is then called an absolute extensor (AE) for normal spaces. One then proves that a space is an (AR) if and only if it is an (AE). Absolute retracts and extensors can be defined for any class of spaces (i.e. not just for normal spaces).
 
Usually an absolute retract (AR) for normal spaces is defined to be a normal space which is a retract of every normal space in which it is imbedded as a closed subset. A space satisfying the property in the article above is then called an absolute extensor (AE) for normal spaces. One then proves that a space is an (AR) if and only if it is an (AE). Absolute retracts and extensors can be defined for any class of spaces (i.e. not just for normal spaces).
  
A binormal space is a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010340/a01034010.png" /> for which the product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010340/a01034011.png" /> is normal. It can be proved that a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010340/a01034012.png" /> is binormal if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010340/a01034013.png" /> is normal and countably paracompact (cf. [[Paracompactness criteria|Paracompactness criteria]]).
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A binormal space is a space $X$ for which the product $X\times I$ is normal. It can be proved that a space $X$ is binormal if and only if $X$ is normal and countably paracompact (cf. [[Paracompactness criteria|Paracompactness criteria]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S.T. Hu,  "Theory of retracts" , Wayne State Univ. Press , Detroit  (1965)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S.T. Hu,  "Theory of retracts" , Wayne State Univ. Press , Detroit  (1965)</TD></TR></table>

Latest revision as of 14:27, 18 August 2014

A topological space $X$ such that every mapping $g\colon A\to X$ of any closed subset $A$ of an arbitrary normal space $Y$ can be extended to the entire space $Y$. A direct product of absolute retracts for normal spaces is an absolute retract for normal spaces, as is any retract (cf. Retract of a topological space) of an absolute retract for normal spaces. In particular, the following spaces are absolute retracts for normal spaces: the unit interval $I$; the $n$-dimensional cube $I^n$; and the Hilbert cube $I^\omega$. Any two mappings of a binormal space into an absolute retract for normal spaces are homotopic; while a binormal absolute retract for normal spaces is contractible into a point.


Comments

Usually an absolute retract (AR) for normal spaces is defined to be a normal space which is a retract of every normal space in which it is imbedded as a closed subset. A space satisfying the property in the article above is then called an absolute extensor (AE) for normal spaces. One then proves that a space is an (AR) if and only if it is an (AE). Absolute retracts and extensors can be defined for any class of spaces (i.e. not just for normal spaces).

A binormal space is a space $X$ for which the product $X\times I$ is normal. It can be proved that a space $X$ is binormal if and only if $X$ is normal and countably paracompact (cf. Paracompactness criteria).

References

[a1] S.T. Hu, "Theory of retracts" , Wayne State Univ. Press , Detroit (1965)
How to Cite This Entry:
Absolute retract for normal spaces. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Absolute_retract_for_normal_spaces&oldid=17595
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article