View source for Absolute retract for normal spaces

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.



====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 <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]]).

====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>