Borel Universal Sets of Compact Spaces
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 which takes a topological space X to a family \Gamma(X) of subsets of X.  For a topological space X, we define U to be a \Gamma-universal set parametrised by a space Y as an element of \Gamma(X×Y) such that for all A in \Gamma(X) there is a y in Y such that A={ x in X: (x, y) in U}.  In the talk, we shall examine \Gamma-universal sets of regular Hausdorff spaces when \Gamma maps to fixed Borel classes.  We shall investigate what properties of X can be inferred from properties of Y if X has such a Borel universal set parametrised by Y.  Special attention will be given to the influence of compactness of X.  We note in particular the following results.

If the compact space X has an open universal set parametrised by Y, then hd(X) <= hd(Y) and hL(X) <= hL(Y).  There is a non-compact space X with an open universal set parametrised by Y such that hd(X) > hd(Y), and there is a consistent example of a non-compact S-space with an open universal set parametrised by an L-space.

Date received: June 30, 1999