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Trichotomies for ideals of compact sets. (English) Zbl 1105.03040
For a Polish space \(E\), \({\mathcal K}(E)\) is the hyperspace of all compact subsets of \(E\) equipped with the Vietoris topology. A family \({\mathcal A}\subset {\mathcal K}(E)\) is rich in sequences at \(x\in E\) if there is a dense set \({\mathcal D}\subset {\mathcal K}(E)\) such that \(\{ x\}\cup \bigcup_n K_n\in {\mathcal A}\) for each sequence \((K_n)_n\subset {\mathcal D}\) convering to \(\{ x\}\). A family \({\mathcal A}\subset {\mathcal K}(E)\) is rich in sequences if \(\bigcup {\mathcal A}\neq\emptyset\) and \(\mathcal A\) is rich in sequences at each point \(x\in\bigcup{\mathcal A}\). The main result of the paper reads as follows. Let \({\mathcal J}\) be a universally Baire ideal of \({\mathcal K}(E)\) which is rich in sequences at some point \(x\in E\). Then either \(\mathcal J\) is a \(\sigma\)-ideal, or \(\mathcal J\) is \(\Pi_3^0\)-hard, or \(\mathcal J\) is \(\Sigma_3^0\)-hard. In the second part of the paper the authors consider connections between richness in sequences and comeagerness of ideals. They prove that if \(\mathcal J\) comeager in \({\mathcal K}(E)\), then \(\mathcal J\) is rich in sequences at comeagerly many points of \(E\). On the other hand, if \(\mathcal J\) is \(\Sigma_3^0\) and rich in sequences at comeagerly many points of \(E\), then \(\mathcal J\) is comeager in \({\mathcal K}(E)\).

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)
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[1] DOI: 10.1007/BF02772948 · Zbl 0769.42006
[2] DOI: 10.5802/aif.1105 · Zbl 0618.42004
[3] Topology and Borel structure 10 (1974)
[4] Colloquium Mathematicum 86 pp 203– (2000)
[5] DOI: 10.1016/0168-0072(94)90029-9 · Zbl 0822.03026
[6] Classical descriptive set theory 156 (1995) · Zbl 0819.04002
[7] DOI: 10.4064/fm185-2-1 · Zbl 1077.03028
[8] Annates de l’Institut Fourier 54 pp 1877– (2004)
[9] Fundamenta Mathematical 158 pp 181– (1998)
[10] Transactions of the American Mathematical Society 301 pp 263– (1987)
[11] Descriptive set theory and the structure of sets of uniqueness 128 (1987) · Zbl 0642.42014
[12] Topics in topology 1652 (1997) · Zbl 0953.54001
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