# Essential mapping

From Encyclopedia of Mathematics

A continuous mapping of a topological space into an open simplex such that every continuous mapping that coincides with at all points of the set is a mapping onto the whole of . For example, the identity mapping of 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 if and only if it admits an essential mapping onto the -dimensional simplex .

#### 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