Souslin scheme

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

A Souslin scheme on $ℱ$ is a collection ${(A_s)}_{s\in 𝒮}$ of sets in $ℱ$.If $𝒩={\mathbb{N}}^{\mathbb{N}}$ is Baire space then, for any $s\in 𝒩$ and $n\in \mathbb{N}$, we write ${s|}_n\equiv (s_1,\mathrm{\dots },s_n)$ for the restriction of $s$ to $\{1,\mathrm{\dots },n\}$. So, ${s|}_n\in {\mathbb{N}}^n$.

The result of the Souslin scheme $(A_s)$ is defined to be

$$A=\bigcup _{s\in 𝒩}\bigcap _{n=1}^{\mathrm{\infty }}{A}_{{s|}_n}.$$

The set $𝒮$ can be partially ordered as follows. Say that $s\le t$ if $s\in {\mathbb{N}}^r$ and $t\in {\mathbb{N}}^s$ for $r\le s$, and $s_k=t_k$ for $k=1,\mathrm{\dots },r$.The scheme $(A_s)$ is said to be regular if $A_s\supseteq A_t$ for all $s\le t$.

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