Suslin representation

In mathematics, a Suslin representation of a set of reals (more precisely, elements of Baire space) is a tree whose projection is that set of reals. More generally, a subset A of κ^ω is λ-Suslin if there is a tree T on κ × λ such that A = p[T].

By a tree on κ × λ we mean here a subset T of the union of κⁱ × λⁱ for all i ∈ N (or i < ω in set-theoretical notation).

Here, p[T] = { f | ∃g : (f,g) ∈ [T] } is the projection of T, where [T] = { (f, g ) | ∀n ∈ ω : (f(n), g(n) ∈ T } is the set of branches through T.

Since [T] is a closed set for the product topology on κ^ω × λ^ω where κ and λ are equipped with the discrete topology (and all closed sets in κ^ω × λ^ω come in this way from some tree on κ × λ), λ-Suslin subsets of κ^ω are projections of closed subsets in κ^ω × λ^ω.

When one talks of Suslin sets without specifying the space, then one usually means Suslin subsets of R, which descriptive set theorists usually take to be the set ω^ω.