# Difference between revisions of "Essential mapping"

From Encyclopedia of Mathematics

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− | A continuous mapping | + | A [[continuous mapping]] $f$ of a [[topological space]] $X$ into an open [[simplex]] $T^n$ such that every continuous mapping $f_1 : X \rightarrow T^n$ that coincides with $f$ at all points of the set $f^{-1}\left({ \bar T^n \ \setminus T^n }\right)$ is a mapping onto the whole of $T^n$. For example, the identity mapping of $T^n$ onto itself is an essential mapping. |

====References==== | ====References==== | ||

− | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> P.S. Aleksandrov, B.A. Pasynkov, "Introduction to dimension theory" , Moscow (1973) (In Russian)</TD></TR></table> | + | <table> |

+ | <TR><TD valign="top">[1]</TD> <TD valign="top"> P.S. Aleksandrov, B.A. Pasynkov, "Introduction to dimension theory" , Moscow (1973) (In Russian)</TD></TR> | ||

+ | </table> | ||

====Comments==== | ====Comments==== | ||

− | Essential mappings are used to characterize the covering dimension (see [[Dimension|Dimension]]) of normal spaces. A [[ | + | Essential mappings are used to characterize the [[covering dimension]] (see [[Dimension|Dimension]]) of normal spaces. A [[normal space]] has covering dimension $\ge n$ if and only if it admits an essential mapping onto the $n$-dimensional simplex $T^n$. |

====References==== | ====References==== | ||

− | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Engelking, "Dimension theory" , North-Holland & PWN (1978)</TD></TR></table> | + | <table> |

+ | <TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Engelking, "Dimension theory" , North-Holland & PWN (1978)</TD></TR> | ||

+ | </table> | ||

+ | |||

+ | {{TEX|done}} | ||

+ | |||

+ | [[Category:General topology]] |

## Latest revision as of 19:51, 3 February 2021

A continuous mapping $f$ of a topological space $X$ into an open simplex $T^n$ such that every continuous mapping $f_1 : X \rightarrow T^n$ that coincides with $f$ at all points of the set $f^{-1}\left({ \bar T^n \ \setminus T^n }\right)$ is a mapping onto the whole of $T^n$. For example, the identity mapping of $T^n$ onto itself is an essential mapping.

#### References

[1] | P.S. Aleksandrov, B.A. Pasynkov, "Introduction to dimension theory" , Moscow (1973) (In Russian) |

#### Comments

Essential mappings are used to characterize the covering dimension (see Dimension) of normal spaces. A normal space has covering dimension $\ge n$ if and only if it admits an essential mapping onto the $n$-dimensional simplex $T^n$.

#### References

[a1] | R. Engelking, "Dimension theory" , North-Holland & PWN (1978) |

**How to Cite This Entry:**

Essential mapping.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Essential_mapping&oldid=12470

This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article