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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.

##### 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
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