Retraction spaces and the homotopy metric

Abstract

AbstractLet X be a finite-dimensional compactum. Let R(X) and N(X) be the spaces of retractions and non-deformation retractions of X, respectively, with the compact-open (=sup-metric) topology. Let 2Xh be the space of non-empty compact ANR subsets of X with topology induced by the homotopy metric. Let RXh be the subspace of 2Xh consisting of the ANR's in X that are retracts of X.We show that N(Sm) is simply-connected for m > 1. We show that if X is an ANR and A0ϵRXh, then limi→∞Ai=A0 in 2Xh if and only if for every retraction r0 of X onto A0 there are, for almost all i, retractions ri of X onto Ai such that limi→∞ri=ro in R(X). We show that if X is an ANR, then the local connectedness of R(X) implies that of RXh. We prove that R(M) is locally connected if M is a closed surface. We give examples to show how some of our results weaken when X is not assumed to be an ANR