# 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 *κ*^{i} × *λ*^{i} 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}.