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Difference between revisions of "Proper cycle"

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''in a metric space''
 
''in a metric space''
  
A sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075440/p0754401.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075440/p0754402.png" />-cycles (cf. [[Vietoris homology|Vietoris homology]]) satisfying the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075440/p0754403.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075440/p0754404.png" />. The compact set on which all vertices of all cycles of all simplices of a proper cycle lie is called the compact support of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075440/p0754405.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075440/p0754406.png" /> is a continuous mapping, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075440/p0754407.png" /> is also a proper cycle, and a deformation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075440/p0754408.png" /> induces a deformation of the proper cycle.
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A sequence $  z  ^ {n} = \{ z _ {1}  ^ {n} , z _ {2}  ^ {n} ,\dots \} $
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of $  \epsilon _ {k} $-
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cycles (cf. [[Vietoris homology|Vietoris homology]]) satisfying the condition $  \epsilon _ {k} \rightarrow 0 $
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as $  k \rightarrow \infty $.  
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The compact set on which all vertices of all cycles of all simplices of a proper cycle lie is called the compact support of $  z $.  
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If $  f : X \rightarrow X $
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is a continuous mapping, then $  f ( z) $
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is also a proper cycle, and a deformation of $  f $
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induces a deformation of the proper cycle.
  
 
Cf. [[Vietoris homology|Vietoris homology]].
 
Cf. [[Vietoris homology|Vietoris homology]].

Latest revision as of 08:08, 6 June 2020


in a metric space

A sequence $ z ^ {n} = \{ z _ {1} ^ {n} , z _ {2} ^ {n} ,\dots \} $ of $ \epsilon _ {k} $- cycles (cf. Vietoris homology) satisfying the condition $ \epsilon _ {k} \rightarrow 0 $ as $ k \rightarrow \infty $. The compact set on which all vertices of all cycles of all simplices of a proper cycle lie is called the compact support of $ z $. If $ f : X \rightarrow X $ is a continuous mapping, then $ f ( z) $ is also a proper cycle, and a deformation of $ f $ induces a deformation of the proper cycle.

Cf. Vietoris homology.

How to Cite This Entry:
Proper cycle. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Proper_cycle&oldid=11895
This article was adapted from an original article by A.A. Mal'tsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article