Generic Souslin sets

Arnold William Miller

Vol. 97 (1981), No. 1, 171–181

DOI: 10.2140/pjm.1981.97.171

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 B_s for s ∈ ω^<ω such that:

For X = 2^ω (the Cantor space) a set Y ⊆ X is analytic iff it is the projection of a Borel subset of 2^ω × 2^ω. 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(2^ω) such that R is countable and every analytic set in 2^ω 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 ⊆ 2^ω}, the abstract rectangles in the plane. Let S(R) be the family of subsets of 2^ω×2^ω obtained by applying the Souslin operation to sets in B(R). The next question was asked by D. Mauldin.

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

   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