Essential 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

[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)