Strongly e-movable convergence and spaces of ANR's

Abstract

Let X be a metric space and let ANR( X) denote the hyperspace of all compact ANR's in X. This paper introduces a notion of a strongly e-movable convergence for sequences in ANR( X) and proves several characterizations of strongly e-movable convergence. For a (complete) separable metric space X we show that ANR( X) with the topology induced by strongly e-movable convergence can be metrized as a (complete) separable metric space. Moreover, if X is a finite-dimensional compactum, then strongly e-movable convergence induces on ANR( X) the same topology as that induced by Borsuk's homotopy metric. For a separable Q-manifold M, ANR( M) is locally arcwise connected and A, B ϵ ANR( M) can be joined by an arc in ANR( M) iff there is a simple homotopy equivalence ϕ: A → B homotopic to the inclusion of A into M.