Souslin scheme


A Souslin scheme is a method of representing and defining analytic setsMathworldPlanetmath on a paved space (X,ℱ). Let 𝒮 be the collectionMathworldPlanetmath of finite sequencesPlanetmathPlanetmath of positive integers. That is 𝒮 is the disjoint unionMathworldPlanetmathPlanetmath of ℕn for n=1,2,….

A Souslin scheme on ℱ is a collection (As)s∈𝒮 of sets in ℱ. If 𝒩=ℕℕ is Baire spacePlanetmathPlanetmath then, for any s∈𝒩 and n∈ℕ, we write s|n≡(s1,…,sn) for the restrictionPlanetmathPlanetmath of s to {1,…,n}. So, s|n∈ℕn.

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

A=⋃s∈𝒩⋂n=1∞As|n.

The set 𝒮 can be partially ordered as follows. Say that s≤t if s∈ℕr and t∈ℕs for r≤s, and sk=tk for k=1,…,r. The scheme (As) is said to be regularPlanetmathPlanetmathPlanetmathPlanetmath if As⊇At for all s≤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).

References

  • 1 Jean Bourgain, A stabilization property and its applications in the theory of sectionsPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath. 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