Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-25T19:01:19.435Z Has data issue: false hasContentIssue false
  • English
  • Français

Cofinal families of Borel equivalence relations and quasiorders

Published online by Cambridge University Press:  12 March 2014

Christian Rosendal
Christian Rosendal*
Affiliation:
Department of Mathematics, University of Illinois at Urbana, 273 Altgeld Hall, MC-382, 1409 West Greet Street, Urbana, IL 61801, USA, E-mail: rosendal@ccr.jussieu.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

Families of Borel equivalence relations and quasiorders that are cofinal with respect to the Borel reducibility ordering. ≤B, are constructed. There is an analytic ideal on ω generating a complete analytic equivalence relation and any Borel equivalence relation reduces to one generated by a Borel ideal. Several Borel equivalence relations, among them Lipschitz isomorphism of compact metric spaces, are shown to be Kσ complete.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 70 , Issue 4 , December 2005 , pp. 1325 - 1340
Copyright
Copyright © Association for Symbolic Logic 2005

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Barwise, J., Back and forth through infinitary logic, Studies in model theory. Mathematical Association of America, 1973.Google Scholar
[2]Bossard, B., A coding of separable Banach spaces. Analytic and coanalytic families of Banach spaces, Fundamenta Mathematicae, vol. 172 (2002), pp. 117–152.CrossRefGoogle Scholar
[3]Clemens, J., Gao, S., and Kechris, A., Polish metric spaces: their classification and isometry groups. The Bulletin of Symbolic Logic, vol. 7 (2001), pp. 361–375.CrossRefGoogle Scholar
[4]Friedman, H., Borel and Baire reducibility, Fundamenta Mathematicae, vol. 164 (2000), pp. 61–69.CrossRefGoogle Scholar
[5]Friedman, H. and Stanley, L., A Borel reducibility theory for classes of countable structures, this JOurnal, vol. 54 (1989), pp. 894–914.Google Scholar
[6]Gao, S. and Kechris, A., On the classification of Polish metric spaces up to isometry, vol. 161. Memoirs of the American Mathematical Society, no. 766, 2003.Google Scholar
[7]Hjorth, G. and Kechris, A., Analytic equivalence relations and Ulm-type classifications, this JOurnal, vol. 60 (1995), pp. 1273–1300.Google Scholar
[8]Kanovei, V., Varia of ideals and ERs, forthcoming book.Google Scholar
[9]Kanovei, V. and Reeken, M., Some new results on Borel irreducibility of equivalence relations, Izvestiya: Mathematics, vol. 67 (2003), no. 1.Google Scholar
[10]Kechris, A., Lectures on definable group actions and equivalence relations, circulated notes (1994).Google Scholar
[11]Kechris, A., Classical descriptive set theory, Springer, 1995.CrossRefGoogle Scholar
[12]Kechris, A. and Louveau, A., The classification of hypersmooth Borel equivalence relations, Journal of the American Mathematical Society, vol. 10 (1997). no. 1, pp. 215–242.CrossRefGoogle Scholar
[13]Louveau, A., Analytic partial orders, preprint.Google Scholar
[14]Louveau, A., On the Borel reducibility ordering, preprint.Google Scholar
[15]Louveau, A. and Rosendal, C., Relations d'equivalence analytiques completes, Comptes Rendus de l'Académie des Sciences. Série I, vol. 333 (2001), pp. 903–906.Google Scholar
[16]Louveau, A. and Rosendal, C., Complete analytic equivalence relations, vol. 357 (2005), pp. 4839–4866.Google Scholar
[17]Rosendal, C., Etude descriptive de l'isomorphisme dans la classe des espaces de Banach, Ph.D. thesis, Université Paris 6, 2003.Google Scholar
[18]Stanley, L., Borel diagonalization and abstract set theory; recent results of Harvey Friedman, Harvey Friedman's research on the foundations of mathematics (Harrington, L.A.et al., editors). North Holland, 1985, pp. 11–86.CrossRefGoogle Scholar