Retraction spaces and the homotopy metric

Let X be a finite-dimensional compactum. Let $\text{R}$(X) and $\text{N}$(X) be the spaces of retractions and non-deformation retractions of X, respectively, with the compact-open (=sup-metric) topology. Let 2^X_h be the space of non-empty compact ANR subsets of X with topology induced by the homotopy metric. Let R^X_h be the subspace of 2^X_h consisting of the ANR's in X that are retracts of X.We show that $\text{N}$(S^m) is simply-connected for m > 1. We show that if X is an ANR and A₀?R^X_h, then lim_i→∞A_i=A₀ in 2^X_h if and only if for every retraction r₀ of X onto A₀ there are, for almost all i, retractions r_i of X onto A_i such that lim_i→∞r_i=r_o in $\text{R}$(X). We show that if X is an ANR, then the local connectedness of $\text{R}$(X) implies that of R^X_h. We prove that $\text{R}$(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.