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 a subset T ⊆ ⋃_n<ω(κⁿ × λⁿ) closed under initial segments, and p[T] = { f∈κ^ω | ∃g∈λ^ω : (f,g) ∈ [T] } is the projection of T, where [T] = { (f, g )∈κ^ω × λ^ω | ∀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.

See also

• Suslin cardinal
• Suslin operation

References

External links

• R. Ketchersid, The strength of an ω₁-dense ideal on ω₁ under CH, 2004.