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, Volume 35, Issue 1, pp 139–155 | Cite as

The Tukey Order and Subsets of ω 1

  • Paul Gartside
  • Ana Mamatelashvili
Article
  • 40 Downloads

Abstract

One partially ordered set, Q, is a Tukey quotient of another, P, if there is a map ϕ : PQ carrying cofinal sets of P to cofinal sets of Q. Two partial orders which are mutual Tukey quotients are said to be Tukey equivalent. Let X be a space and denote by \(\mathcal {K}(X)\) the set of compact subsets of X, ordered by inclusion. The principal object of this paper is to analyze the Tukey equivalence classes of \(\mathcal {K}(S)\) corresponding to various subspaces S of ω 1, their Tukey invariants, and hence the Tukey relations between them. It is shown that ω ω is a strict Tukey quotient of \({\Sigma }(\omega ^{\omega _{1}})\) and thus we distinguish between two Tukey classes out of Isbell’s ten partially ordered sets from (Isbell, J. R.: J. London Math Society 4(2), 394–416, 1972). The relationships between Tukey equivalence classes of \(\mathcal {K}(S)\), where S is a subspace of ω 1, and \(\mathcal {K}(M)\), where M is a separable metrizable space, are revealed. Applications are given to function spaces.

Keywords

Tukey order Compact set Partial order Subsets of ω1 Stationary Separable metrizable space 

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Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of PittsburghPittsburghUSA
  2. 2.Department of Mathematics and StatisticsAuburn UniversityAuburnUSA

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