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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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