# Difference between revisions of "Absolute retract for normal spaces"

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− | A topological space | + | {{TEX|done}} |

+ | 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 | + | 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 12:27, 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://encyclopediaofmath.org/index.php?title=Absolute_retract_for_normal_spaces&oldid=32989