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Open Universal Sets
by
Joseph T H Lo
St Edmund Hall, University of Oxford
Coauthors: Paul M Gartside (University of Oxford)
Let X and Y be topological spaces. The space X is said to have an open universal set U parametrised by Y, if for each open set A of X there is a y in Y such that A = {x in X : (x, y) in U}. All spaces are assumed to be regular Hausdorff. Every space has an open universal set parametrised by its Vietoris hyperspace. We shall note in particular the following two theorems concerning open universal sets.
Suppose X has an open universal set parametrised by Y. Then hd(X) <= hL(Y), hL(X) <= hd(Y).
It is consistent and independent with ZFC that every compact zero-dimensional space with an open universal set parametrised by a hereditarily c.c.c. space is metrisable.
Date received: January 24, 2000
Copyright © 2000 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 # cady-07.