Proper cycle

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.