Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-27T21:50:17.287Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Get access

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

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, 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