The Journal of Symbolic Logic

Research Article

Separation principles and the axiom of determinateness  1

Robert A. van Wesep

University of California, Berkeley, Berkeley, California 94720

California Institute of Technology, Pasadena, California 91125

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.

(Received December 20 1976)


1   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.