# The Tukey Order and Subsets of ω 1

The Tukey Order and Subsets of ω 1 One partially ordered set, Q, is a Tukey quotient of another, P, if there is a map ϕ : P → Q 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 K ( X ) $\mathcal {K}(X)$ the set of compact subsets of X, ordered by inclusion. The principal object of this paper is to analyze the Tukey equivalence classes of K ( S ) $\mathcal {K}(S)$ 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 Σ ( ω ω 1 ) ${\Sigma }(\omega ^{\omega _{1}})$ 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 K ( S ) $\mathcal {K}(S)$ , where S is a subspace of ω 1, and K ( M ) $\mathcal {K}(M)$ , where M is a separable metrizable space, are revealed. Applications are given to function spaces. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Order Springer Journals

# The Tukey Order and Subsets of ω 1

, Volume 35 (1) – Feb 21, 2017
17 pages

/lp/springer_journal/the-tukey-order-and-subsets-of-1-18fN1IDD1j
Publisher
Springer Netherlands
Subject
Mathematics; Geometry; Number Theory
ISSN
0167-8094
eISSN
1572-9273
D.O.I.
10.1007/s11083-017-9423-6
Publisher site
See Article on Publisher Site

### Abstract

One partially ordered set, Q, is a Tukey quotient of another, P, if there is a map ϕ : P → Q 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 K ( X ) $\mathcal {K}(X)$ the set of compact subsets of X, ordered by inclusion. The principal object of this paper is to analyze the Tukey equivalence classes of K ( S ) $\mathcal {K}(S)$ 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 Σ ( ω ω 1 ) ${\Sigma }(\omega ^{\omega _{1}})$ 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 K ( S ) $\mathcal {K}(S)$ , where S is a subspace of ω 1, and K ( M ) $\mathcal {K}(M)$ , where M is a separable metrizable space, are revealed. Applications are given to function spaces.

### Journal

OrderSpringer Journals

Published: Feb 21, 2017

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