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 Σ_{1}^{1} 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 σalgebra 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.
 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.
 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.
 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.
