×

zbMATH — the first resource for mathematics

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.

MSC:
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
PDF BibTeX XML Cite
Full Text: EMIS