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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 κ is λ-Suslin if there is a tree T on κ × λ such that A = p.
By a tree on κ × λ we mean a subset T ⊆ ⋃n<ω(κ × λ) closed under initial segments, and p = { f∈κ | ∃g∈λ : (f,g) ∈ } is the projection of T, where = { (f, g )∈κ × λ | ∀n < ω : (f |n, g |n) ∈ T } is the set of branches through T.
Since is a closed set for the product topology on κ × λ where κ and λ are equipped with the discrete topology (and all closed sets in κ × λ come in this way from some tree on κ × λ), λ-Suslin subsets of κ are projections of closed subsets in κ × λ.
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.
See also
External links
- R. Ketchersid, The strength of an ω1-dense ideal on ω1 under CH, 2004.
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