Parametrizing open universals

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. etrizable 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). 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.