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Separation principles and the axiom of determinateness1

  • Robert A. van Wesep (a1) (a2)

Let Γ be a class of subsets of Baire space (ωω) closed under inverse images by continuous functions. We say such a Γ is continuously closed. Let , the class dual to Γ, consist of the complements relative to ωω of members of Γ. If Γ is not selfdual, i.e., , then let . A continuously closed nonselfdual class Γ of subsets of ωΓ is said to have the first separation property [2] if

The set C is said to separate A and B. The class Γ is said to have the second separation property [3] if

We shall assume the axiom of determinateness and show that if Γ is a continuously closed class of subsets of ωω and then

(1) Γ has the first separation property iff does not have the second separation property, and

(2) either Γ or has the second separation property.

Of course, (1) and (2) taken together imply that Γ and cannot both have the first separation property.

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This paper represents research done at the University of California, Berkeley, under the guidance of Professor J. W. Addison. It is part of the author's Ph.D. thesis. The author wishes to thank Professor Addison for his guidance and encouragement.

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[1]Kuratowski C., Sur les théorèmes de séparation dans la théorie des ensembles, Fundamenta Mathematicae, vol. 26 (1936), pp. 183191.
[2]Lusin N., Sur les ensembles analytiques, Fundamenta Mathematicae, vol. 10 (1927).
[3]Lusin N., Leçons sur les ensembles analytiques et leurs applications, Gauthier-Villars, Paris, 1930.
[4]Martin D. A., The Wadge ordering is a well-ordering, unpublished manuscript.
[5]Steel J., Subsystems of analysis and the axiom of determinateness. Thesis, University of California, Berkeley, 1976.
[6]Van Wesep R., Subsystems of analysis, and descriptive set theory under the axiom of determinateness, Thesis, University of California, Berkeley, 1977.
[7]Wadge W., Degrees of complexity of subsets of the Baire space, Notices of the American Mathematical Society, vol. 19 (1972), p. 714.
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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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