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Proper cycle

From Encyclopedia of Mathematics
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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