Suslin representation

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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 κω 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 iN (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 κω × λω (where κ and λ are equipped with the discrete topology) (and all closed sets in κω × λω come in this way from some tree on κ × λ), λ-Suslin subsets of κω are projections of closed subsets in κω × λω.

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 ωω.

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