RETRACTION SPACES AND THE HOMOTOPY METRIC

Let 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 2(h)(X) he the space of non-empty compact ANR subsets of X with topology induced by the homotopy metric. Let R(h)(X) be the subspace of 2(h)(X) consisting of the ANR's in X that are retracts of X. We show that N(S(m)) is simply-connected for m > 1. We show that if X is an ANR and A(0) is an element of R(h)(X), then lim(i ->infinity) A(1) = A(0) in 2(h)(X) and only if for every retraction r(0) of X onto A(0) there are, for almost all i, retractions r(i) of X onto A(i) such that lim(i ->infinity) r(i) = r(0) in R(X). We show that if X is an ANR, then the local connectedness of R(X) implies that of R(h)(X). 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.