Essential mapping

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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.


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


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 .


[a1] R. Engelking, "Dimension theory" , North-Holland & PWN (1978)
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Essential mapping. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article