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Difference between revisions of "Essential mapping"

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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.
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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 $f^{-1)(\bar T^n \ setminus T^n)$ 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>
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<table>
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<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>
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</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 <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" />.
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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 &amp; PWN  (1978)</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Engelking,  "Dimension theory" , North-Holland &amp; PWN  (1978)</TD></TR>
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</table>
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{{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=12470
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article