Vol. 97, No. 1, 1981

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ISSN: 0030-8730
Generic Souslin sets

Arnold William Miller

Vol. 97 (1981), No. 1, 171–181
Abstract

sforcing we create generic Souslin sets, which we use to answer questions of Ulam, Hansell, and Mauldin. For X a topological space a set Y X is analytic in X(also called Souslin in X or Σ11 in X) iff there are Borel sets Bs for s ω such that:

     ⋃  ⋂
Y =        Bf↾n.
f∈ωωn<ω

For X = 2ω (the Cantor space) a set Y X is analytic iff it is the projection of a Borel subset of 2ω × 2ω. Given R P(X) (the power set of X) let B(R) be the smallest family of subsets of X including R and closed under countable union and complementation (i.e., the σ-algebra generated by R). If X is a topological space and R the family of open sets then B(R) is the family of Borel subsets of X. The following question was raised by Ulam.

  1. Does there exist R P(2ω) such that R is countable and every analytic set in 2ω is an element of B(R)?

    Rothberger showed that assuming CH there is such a R. We will show that it is consistent with ZFC that there is no such R.

  2. Does there exist a separable metric space X in which every subset is analytic but not every subset is Borel?

    This was raised by R. W. Hansell. Clearly CH implies no such X exists. We show that it is consistent with ZFC that such a X exists.

    Let R = {A × B : A,B 2ω}, the abstract rectangles in the plane. Let S(R) be the family of subsets of 2ω×2ω obtained by applying the Souslin operation to sets in B(R). The next question was asked by D. Mauldin.

  3. Does S(R) = P(2ω × 2ω) imply B(R) = P(2ω × 2ω)?

    We show that the answer to this question is no.

Mathematical Subject Classification 2000
Primary: 03E35
Secondary: 03E15, 54H05
Milestones
Received: 18 February 1980
Revised: 28 July 1980
Published: 1 November 1981
Authors
Arnold William Miller