zbMATH — the first resource for mathematics

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 \({\mathcal K}(E)\).

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)
Full Text: DOI
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.