A new notion of \(\kappa\)-Souslin operation. (English) Zbl 0681.04002
I describe, for each infinite cardinal \(\kappa\), an operator \(S_{\kappa}\) with the properties (i) for any family \({\mathcal A}\) of sets, \({\mathcal A}\subseteq S_{\kappa}({\mathcal A})=S_{\kappa}(S_{\kappa}({\mathcal A}))\) and \(S_{\kappa}({\mathcal A})\) is closed under \(\kappa\)-unions and \(\kappa\)-intersections, (ii) if (X,\({\mathcal T})\) is a topological space of weight at most \(\kappa\), and \({\mathcal H}\) is the family of open sets in \(\kappa^{\kappa}\times X\), then there is a set \(B\in S_{\kappa}({\mathcal H})\) such that \(S_{\kappa}({\mathcal T})\) is precisely the set of vertical sections of B, (iii) if (X,\(\Sigma\),\(\mu)\) is a complete \(\kappa^+\)- additive probability space, then \(S_{\kappa}(\Sigma)=\Sigma\), (iv) \(S_{\omega}\) is the usual Souslin operation S; generally \(S_{\kappa}\) has many of the properties one seeks in a generalization of S. (The particular advance I claim is that \(S_{\kappa}({\mathcal A})\) is always closed under \(\kappa\)-intersections, not just countable intersections.) I give applications in measure theory and general topology.
Reviewer: D.H.Fremlin
MSC:
03E15 | Descriptive set theory |
28A05 | Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets |
54H05 | Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets) |
Keywords:
Souslin operationReferences:
[1] | Stone, Analytic sets in non-separable metric spaces pp 471– |
[2] | Souslin, C.R. Acad. Sci. (Paris) 164 pp 88– (1917) |
[3] | Rogers, Analytic Sets (1980) |
[4] | Fremlin, Consequences of Martin’s Axiom (1984) · Zbl 0551.03033 · doi:10.1017/CBO9780511896972 |
[5] | Lusin, C.R. Acad. Sci. (Paris) 164 pp 91– (1917) |
[6] | Kuratowski, Topology I (1966) |
[7] | Fremlin, Dissertationes Math 260 pp 1– (1987) |
[8] | Moschovakis, Descriptive Set Theory (1980) |
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