Parametrizing open universals

Abstract

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 parametrized by Y if for each open set V of X, there is y∈ Y such that V={x∈X: (x,y)∈U} . A space Y is said to parametrize W(κ) if Y parametrizes an open universal set of each space of weight less than or equal to κ. The following are the important results of this paper. If a metrizable space of weight κ parametrizes W(κ) , then κ has countable cofinality. If κ is a strong limit of countable cofinality, then there is a metrizable space of weight κ parametrizing W(κ) . It is consistent and independent that there is a cardinal κ of countable cofinality, but not a strong limit, and a metrizable space of weight κ parametrizing W(κ) . It is consistent and independent that a zero-dimensional, compact first countable space parametrizing itself (equivalently, parametrizing all spaces of the same or smaller weight) must be metrizable.