Difference between revisions of "Essential mapping"
From Encyclopedia of Mathematics
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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. | + | 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. |
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| − | Essential mappings are used to characterize the covering dimension (see [[Dimension|Dimension]]) of normal spaces. A [[ | + | Essential mappings are used to characterize the [[covering dimension]] (see [[Dimension|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 $T^n$. |
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Latest revision as of 19:51, 3 February 2021
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=34344
Essential mapping. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Essential_mapping&oldid=34344
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article