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かっぱ^ωおめが is λらむだ-Suslin if there is a tree T on κかっぱ × λらむだ such that A = p[T].

By a tree on κかっぱ × λらむだ we mean a subset T ⊆ ⋃_{n<ωおめが}(κかっぱⁿ × λらむだⁿ) closed under initial segments, and p[T] = { f∈κかっぱ^ωおめが | ∃g∈λらむだ^ωおめが : (f,g) ∈ [T] } is the projection of T, where [T] = { (f, g )∈κかっぱ^ωおめが × λらむだ^ωおめが | ∀n < ωおめが : (f |n, g |n) ∈ T } is the set of branches through T.

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

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おめが^ωおめが.

See also

• Suslin cardinal
• Suslin operation

External links

• R. Ketchersid, The strength of an ωおめが₁-dense ideal on ωおめが₁ under CH, 2004.