Shape as a Cantor completion process

Abstract

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{ε>0:f is homotopic to g in the ε-neighborhood of Y}, and a sequence of maps fk :X→Q, k∈N, is said to be a Cauchy sequence provided for every ε>0 there is a k0∈N such that F(fk,fk′)<ε 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(α,β):=lim k F(fk,gk), where the shape morphisms α,β∈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.