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Parametrizing open universals

Authors
Journal
Topology and its Applications
0166-8641
Publisher
Elsevier
Publication Date
Volume
119
Issue
2
Identifiers
DOI: 10.1016/s0166-8641(01)00069-4
Keywords
  • Open Universal Set
  • Metrizable Space
  • Cantor Cube (Generalised)
  • Bernstein Set (Generalised)
  • σ-Weight
Disciplines
  • Mathematics

Abstract

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.

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