All spaces are assumed to be regular Hausdorff topological spaces. If X and Y are spaces, then an open set U in X × Y is an open universal set parametrised by Y if each open set of X is of the form {x in X : (x,y) in U}. A space Y is said to parametrise W(\kappa) if Y parametrises an open universal set of each space of weight less than or equal to \kappa.
If a metrisable space of weight \kappa parametrises W(\kappa), then \kappa has countable cofinality. If \kappa is a strong limit of countable cofinality, then there is a metrisable space of weight \kappa parametrising W(\kappa). It is consistent and independent that there is a cardinal \kappa of countable cofinality, but not a strong limit, and a metrisable space of weight \kappa parametrising W(\kappa).
It is consistent and independent that a zero-dimensional, compact first countable space parametrising itself (equivalently, parametrising all spaces of the same or smaller weight) must be metrisable.
Keywords: Open universal set, metrisable space, compact space,
(generalised) Cantor cube, (generalised) Bernstein set, \lambda-weight,
G\delta-diagonals
MCS: 54A25, 04A20, 54E35, Secondary 54A35, 54D65, 54D30, 54D15
This article was published. Gartside, Paul M.; Knight, Robin W.; Lo, Joseph T.H. Parametrizing open universals. Topology and its Applications, Vol: 119, Issue: 2, pp. 131-145; PII S0166-8641(01)00069-4
Date received: September 24, 1999.