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Avoiding families and Tukey functions on the nowhere-dense ideal

Part of: Set theory

Published online by Cambridge University Press:  01 September 2010

Sławomir Solecki  and
Stevo Todorcevic
Sławomir Solecki
Affiliation:
Department of Mathematics, University of Illinois, 1409 W. Green Street, Urbana, IL 61801, USA (ssolecki@math.uiuc.edu)
Stevo Todorcevic
Affiliation:
Université Paris 7-CNRS, FRE 3233, 2, place Jussieu, 75251 Paris Cedex 05, France (stevo@math.jussieu.fr) and Department of Mathematics, University of Toronto, 40 St. George Street, Toronto M5S 2E4, Canada
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Abstract

We investigate Tukey functions from the ideal of all closed nowhere-dense subsets of 2ℕ. In particular, we answer an old question of Isbell and Fremlin by showing that this ideal is not Tukey reducible to the ideal of density zero subsets of ℕ. We also prove non-existence of various special types of Tukey reductions from the nowhere-dense ideal to analytic P-ideals. In connection with these results, we study families of clopen subsets of 2ℕ with the property that for each nowhere-dense subset of 2ℕ there is a set in not intersecting it. We call such families avoiding.

Keywords

Tukey reductionsnowhere-dense setsanalytic P-ideals

MSC classification

Secondary: 03E05: Other combinatorial set theory 03E15: Descriptive set theory
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 10 , Issue 2 , April 2011 , pp. 405 - 435
Copyright
Copyright © Cambridge University Press 2010

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