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Strong expansions and strong shape theory. (English) Zbl 0715.55008

The ANR-approach to shape theory had in a sense its starting point in the fact that a compactum is homeomorphic to the inverse limit of ANR’s (polyhedra) (an expansion of a compactum into the inverse system). The development of shape theory has brought notions as ANR-expansion in the sense of K. Morita [Fundam. Math. 86, 251-259 (1975; Zbl 0296.54034)], ANR-resolution of the author [Fundam. Math. 114, 53-78 (1981; Zbl 0411.54019)] coherent expansion and some others. Here, the notion of strong expansion of a space is introduced and studied, and in particular, the strong ANR-expansions are applied to find the strong shape category. The notion itself is an intermediate notion between resolution of a space and expansion in the sense of K. Morita. It is a stronger form of Morita’s expansion in the sense that the homotopies postulated in condition (iii) of Morita’s [loc. cit.] definition of expansion are related via a certain homotopy. Since this additional homotopy is naturally postulated its output is such that the strong ANR- expansions can be used to construct the strong shape category based on this notion. This is obtained by proving that every strong expansion of a space X is a coherent expansion of X, and then results are applied obtained by Yu. T. Lisitsa and the author [Glas. Mat., III. Ser. 19(39), 335-399 (1984; Zbl 0553.55009)] on coherent expansions in order to define the strong shape.
Reviewer: I.Ivanšić

MSC:

55P55 Shape theory
54C56 Shape theory in general topology
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