# Souslin scheme

A Souslin scheme is a method of representing and defining analytic sets on a paved space $(X,\mathcal{F})$. Let $\mathcal{S}$ be the collection of finite sequences of positive integers. That is $\mathcal{S}$ is the disjoint union of $\mathbb{N}^{n}$ for $n=1,2,\ldots$.

A Souslin scheme on $\mathcal{F}$ is a collection $(A_{s})_{s\in\mathcal{S}}$ of sets in $\mathcal{F}$. If $\mathcal{N}=\mathbb{N}^{\mathbb{N}}$ is Baire space then, for any $s\in\mathcal{N}$ and $n\in\mathbb{N}$, we write $s|_{n}\equiv(s_{1},\ldots,s_{n})$ for the restriction of $s$ to $\{1,\ldots,n\}$. So, $s|_{n}\in\mathbb{N}^{n}$.

The result of the Souslin scheme $(A_{s})$ is defined to be

 $A=\bigcup_{s\in\mathcal{N}}\bigcap_{n=1}^{\infty}A_{s|_{n}}.$

The set $\mathcal{S}$ can be partially ordered as follows. Say that $s\leq t$ if $s\in\mathbb{N}^{r}$ and $t\in\mathbb{N}^{s}$ for $r\leq s$, and $s_{k}=t_{k}$ for $k=1,\ldots,r$. The scheme $(A_{s})$ is said to be regular if $A_{s}\supseteq A_{t}$ for all $s\leq t$.

It can be shown that the result of a Souslin scheme is $\mathcal{F}$-analytic and, conversely, any analytic set is the result of some scheme (see equivalent definitions of analytic sets).

## References

• 1 Jean Bourgain, A stabilization property and its applications in the theory of sections. Séminaire Choquet. Initiation à l’analyse, 17 no. 1 (1977).
Title Souslin scheme SouslinScheme 2013-03-22 18:48:30 2013-03-22 18:48:30 gel (22282) gel (22282) 5 gel (22282) Definition msc 28A05 Suslin scheme regular scheme result of a scheme