Hostname: page-component-77c89778f8-m42fx Total loading time: 0 Render date: 2024-07-21T11:06:21.991Z Has data issue: false hasContentIssue false
  • English
  • Français

Two results on borel orders

Published online by Cambridge University Press:  12 March 2014

Alain Louveau
Alain Louveau*
Affiliation:
Équipe d'Analyse, Université Paris-VI et CNRS, 75252 Paris, France
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove two results about the embeddability relation between Borel linear orders: For η a countable ordinal, let 2η (resp. 2< η) be the set of sequences of zeros and ones of length η (resp. < η), equipped with the lexicographic ordering. Given a Borel linear order X and a countable ordinal ξ, we prove the following two facts.

(a) Either X can be embedded (in a (X, ξ) way) in 2ωξ or 2ωξ + 1 continuously embeds in X.

(b) Either X can embedded (in a (X, ξ) way) in 2<ωξ or 2ωξ continuously embeds in X. These results extend previous work of Harrington, Shelah and Marker.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 54 , Issue 3 , September 1989 , pp. 865 - 874
Copyright
Copyright © Association for Symbolic Logic 1989

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

[F]Friedman, H., Borel structures in mathematics, manuscript, Ohio State University, Columbus, Ohio, 1979.Google Scholar
[HMS]Harrington, L., Marker, D., and Shelah, S., Borel orderings, Transactions of the American Mathematical Society, vol. 310 (1988), pp. 293–302.CrossRefGoogle Scholar
[HS]Harrington, L. and Shelah, S., Counting equivalence classes for co-κ-Suslin equivalence relations, Logic colloquium '80, North-Holland, Amsterdam, 1982, pp. 147–152.Google Scholar
[S]Shelah, S., On co-κ-Suslin relations, Israel Journal of Mathematics, vol. 47 (1984), pp. 139–153.CrossRefGoogle Scholar