Difference between revisions of "Essential mapping"
From Encyclopedia of Mathematics
(Importing text file) |
(LaTeX) |
||
Line 1: | Line 1: | ||
− | A continuous mapping | + | A continuous mapping 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)(\bar T^n \ setminus T^n) is a mapping onto the whole of T^n. For example, the identity mapping of T^n onto itself is an essential mapping. |
====References==== | ====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> | + | <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==== | ====Comments==== | ||
− | Essential mappings are used to characterize the covering dimension (see [[Dimension|Dimension]]) of normal spaces. A [[Normal space|normal space]] has covering dimension | + | Essential mappings are used to characterize the covering dimension (see [[Dimension|Dimension]]) of normal spaces. A [[Normal space|normal space]] has covering dimension \ge n if and only if it admits an essential mapping onto the n-dimensional simplex T^n. |
====References==== | ====References==== | ||
− | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Engelking, "Dimension theory" , North-Holland & PWN (1978)</TD></TR></table> | + | <table> |
+ | <TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Engelking, "Dimension theory" , North-Holland & PWN (1978)</TD></TR> | ||
+ | </table> | ||
+ | |||
+ | {{TEX|done}} |
Revision as of 08:08, 8 November 2014
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)(\bar T^n \ setminus T^n) 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=34342
Essential mapping. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Essential_mapping&oldid=34342
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article