Spaces of Aanr's

Abstract

For a metric space X, let AANRC(X) denote the hyperspace of all nonempty approximative absolute neighborhood retracts in the sense of Clapp in X topologized with the metric of continuity. We show that AANRC(X) is topologically complete iff X is topologically complete. Some subsets of the first Baire category in AANRc(X) for a Q-manifold X are identified. For example, the collection AANRN(X) of all nonempty approximative absolute neighborhood retracts in the sense of Noguchi in X is such a subset. Introduction. In 1953 Noguchi [N] introduced a generalization of the notion of an absolute neighborhood retract (ANR) which is now called an approximative absolute neighborhood retract in the sense of Noguchi (AANRN). A further generalization was given in 1971 by Clapp [C] and is now known as an approximative absolute neighborhood retract in the sense of Clapp (AANRc). The importance of these generalizations is that they share many properties with ANR's (for example, certain fixed point properties). On the other side, they include compacta with local pathologies. Recently, several authors studied AANRc's using techniques of shape theory. In particular, Borsuk [B2] described them as NE-sets, Mardegic [M] characterized them as approximate polyhedra, and the author [C2] observed that AANRk's coincide with P-e-movable compacta, where P denotes the class of all finite polyhedra. The main results of this paper, described in the above abstract, are obtained in the following way. First we define the notion of a P-e-movably regular convergence for compacta in a metric space X. The limit of a P-e-movably regularly convergent sequence must be P-e-movable (i.e., an AANRC) so that a sequence {An} in AANRc(X) converges P-e-movably regularly to AO E AANRC(X) iff lim dc(An, AO) = 0, where dc is Borsuk's metric of continuity [B1]. Then we apply the method of investigating topological properties of the hyperspace of all P-movable compacta in X with the topology induced by P-movably regular convergence from [C1] to AANRc(X) using P-e-movably regular convergence. Hence, in essence, our proof of the statement about topological completeness relies on Begle's method in [Be]. Received by the editors May 16, 1980. AMS (MOS) subject classifications (1970). Primary 54B20, 54B55, 54F40.