Borel Universal Sets
by
Joseph T H Lo
St Edmund Hall, University of Oxford
Coauthors: Paul M Gartside (Merton College, University of Oxford)

Let \Gamma be a class function mapping a topological space X to a family \Gamma(X) of subsets of X. A subset U of X ×Y is a \Gamma-universal set for X parametrised by Y if U in \Gamma(X ×Y) and for each A in \Gamma(X) there is a y in Y such that A=U^y, where U^y={x in X: (x, y) in U }.

We will consider how the properties of the parametrising space Y affect those of the space X, in the case when \Gamma(X) is a Borel class of X. Example results include:

Theorem If X has an open universal parametrised by a second countable space, then X is metrisable.

Example There is a non-metrisable space with a G_\delta-universal parametrised by the Cantor set.

Theorem If compact X has a \Sigma_n⁰ -universal, for n in \omega, parametrised by a second countable space, then X is metrisable.

Example If there is a Q-set or if holds then there is a compact non-metrisable space with a \Sigma⁰_\omega-universal parametrised by the Cantor set.

(All spaces are T₃.)

Date received: May 26, 1999

Copyright © 1999 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 # cacl-25.