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 $n=1,2,\mathrm{\dots}$.
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 ${s|}_{n}\equiv ({s}_{1},\mathrm{\dots},{s}_{n})$ 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
$$A=\bigcup _{s\in \mathcal{N}}\bigcap _{n=1}^{\mathrm{\infty}}{A}_{{s|}_{n}}.$$ |
The set $\mathcal{S}$ 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 $\mathcal{F}$-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 |
---|---|
Canonical name | SouslinScheme |
Date of creation | 2013-03-22 18:48:30 |
Last modified on | 2013-03-22 18:48:30 |
Owner | gel (22282) |
Last modified by | gel (22282) |
Numerical id | 5 |
Author | gel (22282) |
Entry type | Definition |
Classification | msc 28A05 |
Synonym | Suslin scheme |
Defines | regular scheme |
Defines | result of a scheme |