Metrizable shape and strong shape equivalences

In this paper we construct a functor $\Phi : \mbox{${\rm pro}\mathcal Top$} \to \mbox{${\rm pro}\mathcal {ANR}$}$ which extends Mardešic correspondence that assigns to every metrizable space its canonical $\mathcal {ANR}$-resolution. Such a functor allows one to define the strong shape category of prospaces and, moreover, to define a class of spaces, called strongly fibered, that play for strong shape equivalences the role that $\mathcal {ANR}$-spaces play for ordinary shape equivalences. In the last section we characterize SSDR-promaps, as defined by Dydak and Nowak, in terms of the strong homotopy extension property considered by the author.

Keywords

metrizable proreflector, shape equivalence, strong shape equivalence, double mapping cylinder, SSDR-promaps, strong homotopy extension property

2010 Mathematics Subject Classification

18B30, 18G55, 54B30, 55P05, 55P10, 55P55, 55P60

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Published 1 January 2002