Difference between revisions of "Proper cycle"
From Encyclopedia of Mathematics
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| + | $#C+1 = 8 : ~/encyclopedia/old_files/data/P075/P.0705440 Proper cycle | ||
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''in a metric space'' | ''in a metric space'' | ||
| − | A sequence | + | A sequence $ z ^ {n} = \{ z _ {1} ^ {n} , z _ {2} ^ {n} ,\dots \} $ |
| + | of $ \epsilon _ {k} $- | ||
| + | cycles (cf. [[Vietoris homology|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|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
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