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

Loading next page...
 
/lp/springer_journal/the-tukey-order-and-subsets-of-1-18fN1IDD1j
Publisher
Springer Journals
Copyright
Copyright © 2017 by Springer Science+Business Media Dordrecht
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

References

You’re reading a free preview. Subscribe to read the entire article.


DeepDyve is your
personal research library

It’s your single place to instantly
discover and read the research
that matters to you.

Enjoy affordable access to
over 18 million articles from more than
15,000 peer-reviewed journals.

All for just $49/month

Explore the DeepDyve Library

Search

Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly

Organize

Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place.

Access

Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals.

Your journals are on DeepDyve

Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more.

All the latest content is available, no embargo periods.

See the journals in your area

DeepDyve

Freelancer

DeepDyve

Pro

Price

FREE

$49/month
$360/year

Save searches from
Google Scholar,
PubMed

Create lists to
organize your research

Export lists, citations

Read DeepDyve articles

Abstract access only

Unlimited access to over
18 million full-text articles

Print

20 pages / month

PDF Discount

20% off