# Essential mapping

From Encyclopedia of Mathematics

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=51515

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