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A non-continuous description of the shape category of compacta. (English) Zbl 0697.55012
J. E. Felt [Proc. Am. Math. Soc. 46, 426-430 (1974; Zbl 0292.55013)] has related \(\epsilon\)-continuous maps and shape morphisms between metric compacta obtaining a bijection between shape morphisms and inverse limits of certain equivalence classes of \(\epsilon\)-continuous functions. In the reviewed article the author does further investigations and defines the notion of a proximate net \(\alpha =(\alpha_ n): X\to Y\) as a sequence of (not necessarily continuous) functions \(\alpha_ n: X\to Y\) such that for every \(\epsilon >0\) there exists an index \(n_ 0\) such that \(\alpha_ n\) is \(\epsilon\)-homotopic to \(\alpha_{n+1}\) for every \(n\geq n_ 0\). The class of metric compacta together with homotopy classes of proximate nets as morphisms is then organized into a category which is shown to be isomorphic to the shape category of metric compacta. The last section concerns internally movable compacta where if the codomain is internally movable then every homotopy class of proximate nets has a continuous representative.
Reviewer: I.Ivanšić

MSC:
55P55 Shape theory
54C56 Shape theory in general topology
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