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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
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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

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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