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On the complexity of some \(\sigma \)-ideals of \(\sigma \)-P-porous sets. (English) Zbl 1099.54029
Summary: Let \(\mathbf P\) be a porosity-like relation on a separable locally compact metric space \(E\). We show that the \(\sigma \)-ideal of compact \(\sigma \)-\(\mathbf P\)-porous subsets of \(E\) (under some general conditions on \(\mathbf P\) and \(E\)) forms a \(\mathbf {\Pi } _{\mathbf 1}^{\mathbf 1}\)-complete set in the hyperspace of all compact subsets of \(E\), in particular it is coanalytic and non-Borel. Our general results are applicable to most interesting types of porosity. This is shown in the cases of the \(\sigma \)-ideals of \(\sigma \)-porous sets, \(\sigma \)-\(\langle g \rangle \)-porous sets, \(\sigma \)-strongly porous sets, \(\sigma \)-symmetrically porous sets and \(\sigma \)-strongly symmetrically porous sets. We prove a similar result also for \(\sigma \)-very porous sets assuming that each singleton of \(E\) is very porous set.

54H05 Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)
28A05 Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets
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