Nonseparable Analytic Metric Spaces and Quotient Maps
by
R. W. Hansell

One characterization of a separable analytic metric space X is the existence of a continuous surjection f : P --> X where P=\omega^\omega is the space of irrationals. In [1] it is shown that the map f can in fact be taken to be an almost-open continuous map. Consequently, the separable analytic metric spaces are precisely the quotients of the space of irrationals (throughout, all quotients maps are assumed to be continuous). Is there something analogous to this for nonseparable analytic metric spaces (i.e., those metric spaces that always embed as a Souslin-F set)? As noted in [1], the proof for the separable case can be generalized to show that, for any infinite cardinal \kappa, any continuous image of \kappa^\omega is also an almost-open continuous image of \kappa^\omega. Since any metric space of cardinality < \kappa is a continuous image of \kappa^\omega, this latter result does not provide the desired analog for nonseparable analytic spaces. We show that the nonseparable analytic metric spaces are precisely the quotient s-images of the closed subspaces of the Baire spaces \kappa^\omega. By an s-image we mean the map has the property that all fibers are separable. In particular, the following is shown.

Theorem For a metric space X of infinite weight <= \kappa the following are equivalent.

1. X is analytic.
2. X is a quotient s-image of a closed subspace of \kappa^\omega.
3. X is a (continuous) almost-open s-image of a closed subspace of \kappa^\omega.
4. X is a (continuous) almost-open s-image of \kappa^\omega under a map which preserves \sigma-discrete sets and is such that \partialf^-1(y) is separable for each y in Y.

In addition, it is shown that a metric quotient s-image of an analytic metric space is again analytic, and examples are given to show why the various assumptions are needed.

1 E. Michael and A. H. Stone Quotients of the space of irrationals 28 1969 629-633

Date received: June 24, 1996

Copyright © 1996 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caah-31.