Essential mapping

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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.


[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 $\ge n$ if and only if it admits an essential mapping onto the $n$-dimensional simplex $T^n$.


[a1] R. Engelking, "Dimension theory" , North-Holland & PWN (1978)
How to Cite This Entry:
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