Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-12-01T01:16:05.113Z Has data issue: false hasContentIssue false

Non-isomorphic projective sets

Published online by Cambridge University Press:  26 February 2010

R. Daniel Mauldin
Affiliation:
University of Florida, Gainesville, Florida 32611
Get access

Extract

It is well known that two Borel subsets of the unit interval are Borel isomorphic, if, and only if, they have the same cardinality. The problem of the existence of analytic, non-Borelian subsets of the unit interval, which are not Borel isomorphic, has not been resolved within ZFC. With the additional assumption of the existence of an uncountable coanalytic set which does not contain a perfect set, it has been shown that there are at least three Borel isomorphism classes of analytic non-Borelian sets [4, 5].

Type
Research Article
Copyright
Copyright © University College London 1976

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

1.Kuratowski, K.. Topology, Volume I (Academic Press, New York, 1966).Google Scholar
2.Kuratowski, C.. “Ensembles projectifs et ensembles singuliers”, Fund. Math., 35 (1948), 131150. MR10, p. 35CrossRefGoogle Scholar
3.Sierpinski, W.. “Sur un probleme concernant les fonctions projectives”, Fund. Math., 30 (1938), 5960.CrossRefGoogle Scholar
4.Maitra, A. and Ryll-Nardzewski, C.. “On the existence of two analytic non–Borel sets which are not isomorphic”, Bull. Acad. Polon. Sci., 18 (1970), 177178. MR42#3743.Google Scholar
5.Mauldin, R. D.. “On non-isomorphic analytic sets”, to appear Proc. Amer. Math. Soc.Google Scholar
6.Godel, K.. “The consistency of the axiom of choice and the generalized continuum hypothesis”, Proc. Nat. Acad. Sci., 24 (1938), 56CrossRefGoogle ScholarPubMed
7.Silver, J. H.Measurable Cardinals and A31 well-orderings”, Annals of Math., 94 (1971), 414446CrossRefGoogle Scholar