On Zero-Dimensional Images of Compact Ordered Spaces
by
Lutz Heindorf
Freie Universität Berlin

Below X denotes a compact and zero-dimensional space and exp X its exponential, i.e. the space of all non-empty closed subsets of X equipped with the Vietoris topology.

A (not necessarily continuous) mapping \Phi: X ×exp X --> X will be called a complete system of retractions for X, if the mapping x --> \Phi(F, x) is a retraction (i.e. continuous and idempotent) of X onto F, for all F in exp X.

Theorem 1 The space X is a continuous image of a compact ordered space iff there is a complete system \Phi of retractions for X, which is monotone, i.e. such that F subset or equal G implies \Phi(\Phi(x, G), F) = \Phi(x, F).

Theorem 2 The space X is metrizable iff there is a continuous complete system of retractions for X.

Both theorems can be proved by first translating the assertions into the language of Boolean algebra. The if-part of Theorem 2 uses Gruenhage's metrization theorem; one establishes that X² \\Delta is paracompact.

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 Mathematical Conference Abstracts. Document # caah-35.