View source for Essential mapping

A continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036250/e0362501.png" /> of a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036250/e0362502.png" /> into an open simplex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036250/e0362503.png" /> such that every continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036250/e0362504.png" /> that coincides with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036250/e0362505.png" /> at all points of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036250/e0362506.png" /> is a mapping onto the whole of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036250/e0362507.png" />. For example, the identity mapping of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036250/e0362508.png" /> onto itself is an essential mapping.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.S. Aleksandrov,   B.A. Pasynkov,   "Introduction to dimension theory" , Moscow  (1973)  (In Russian)</TD></TR></table>



====Comments====
Essential mappings are used to characterize the covering dimension (see [[Dimension|Dimension]]) of normal spaces. A [[Normal space|normal space]] has covering dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036250/e0362509.png" /> if and only if it admits an essential mapping onto the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036250/e03625010.png" />-dimensional simplex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036250/e03625011.png" />.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Engelking,   "Dimension theory" , North-Holland &amp; PWN  (1978)</TD></TR></table>