Resumen:
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Let X and Y be metric compacta, Y embedded in the Hilbert cube Q. For two maps f,g:X?Q the authors define F(f,g):=inf{?>0:f is homotopic to g in the ?-neighborhood of Y}, and a sequence of maps fk :X?Q, k?N, is said to be a Cauchy sequence provided for every ?>0 there is a k0?N such that F(fk,fk?) whenever k,k??k 0. Such sequences coincide with the approximative maps of K. Borsuk [Theory of shape, PWN, Warsaw, 1975] and represent shape morphisms from X to Y. The function F is not a pseudometric, but defining d(?,?):=lim k F(fk,gk), where the shape morphisms ?,??Sh(X,Y) are represented by Cauchy sequences (fk),(gk), the authors prove that (Sh(X,Y),d) becomes a complete zero-dimensional ultrametric space, homeomorphic to a closed subset of the irrationals. Among other things, the authors prove that if two compacta X and Y are of the same shape, then for every compactum Z, the spaces Sh?(X,Z) and Sh?(Y,Z) are uniformly homeomorphic. In the last section, the authors show, for example, that for X compact and Y?FANR , the space Sh?(X,Y) is countable, give several characterizations of various kinds of movability, and translate their results to Z-sets in Q and sequences of proper maps between their complements.
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