Counting shape and homotopy types among fundamental absolute neighborhood retracts - an elementary approach

Abstract

The authors deal with the question of whether the sets of shape and homotopy types of FANRs are countable. FANRs can be considered as a natural generalization of ANRs; thus the authors' starting point consists of results on the countability of the set of homotopy types of compact metric ANRs proven by M. Mather [Topology 4 (1965), 93-94], by J. Cheeger and J. Kister [Topology 9 (1970), 149-151], and by Kister [Proc. Amer. Math. Soc. 19 (1968), 195]. Because of the fact that FANRs do not have in general the shape of finite polyhedra, these results cannot be applied to the situation of FANRs. The authors prove that the set of shape types of FANRs is countable, but that the set of homotopy types of spaces shape equivalent to a compact X is not countable. As a consequence they show that every FANR can be embedded up to shape as a retract of a movable compactum, and can be topologically embedded as a shape retract of a compactum.