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A note on universal sets for classes of countable Gδ'S

Published online by Cambridge University Press:  26 February 2010

A. S. Kechris  and
D. A. Martin
A. S. Kechris
Affiliation:
Massachusetts Institute of Technology, California Institute of Technology, The Rockefeller University.
D. A. Martin
Affiliation:
Massachusetts Institute of Technology, California Institute of Technology, The Rockefeller University.
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Extract

In a recent article [2] D. G. Larman and C. A. Rogers proved the following two results in Descriptive Set Theory (where R = the space of real numbers): (1) There is no analytic set in the plane R2, which is universal for the countable closed subsets of R; (2) there is no Borel set in R2, which is universal for the countable Gδ subsets of R. Recall that, if b is a class of subsets of a space X, a set U ⊆ X × X is called universal for b if (a) for each x ∈ X, Ux = def {y : (x, y) ∈ U} ∈ b, and (b) for each A ∈ b there is an x such that A = Ux. (Larman and Rogers have also shown that in both cases coanalytic universal sets exist.)

Type
Research Article
Information
Mathematika , Volume 22 , Issue 1 , June 1975 , pp. 43 - 45
Copyright
Copyright © University College London 1975

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References

1. Kuratowski, K.. Topology, vol. 1 (Academic Press, New York & London, 1966).Google Scholar
2. Larman, D. G. and Rogers, C. A.. “The Descriptive Character of Certain Universal sets”, Proc. London Math. Soc, (3), 27 (1973), 385–401.CrossRefGoogle Scholar