Geodesic
On the complexity of some $\sigma$-ideals of $\sigma$-P-porous sets
Commentationes Mathematicae Universitatis Carolinae, Tome 44 (2003) no. 3, pp. 531-554 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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Let $\bold P$ be a porosity-like relation on a separable locally compact metric space $E$. We show that the $\sigma$-ideal of compact $\sigma$-$\bold P$-porous subsets of $E$ (under some general conditions on $\bold P$ and $E$) forms a $\boldsymbol \Pi_{\bold 1}^{\bold 1}$-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. It 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.
Let $\bold P$ be a porosity-like relation on a separable locally compact metric space $E$. We show that the $\sigma$-ideal of compact $\sigma$-$\bold P$-porous subsets of $E$ (under some general conditions on $\bold P$ and $E$) forms a $\boldsymbol \Pi_{\bold 1}^{\bold 1}$-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. It 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.
Classification : 28A05, 54H05, 54H25
Keywords: $\sigma $-porous sets; $\sigma $-ideal; coanalytic sets; Hausdorff metric 
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     title = {On the complexity of some $\sigma$-ideals of $\sigma${-P-porous} sets},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {531--554},
     year = {2003},
     volume = {44},
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     zbl = {1099.54029},
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     url = {http://geodesic.mathdoc.fr/item/CMUC_2003_44_3_a11/}
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Zajíček, Luděk; Zelený, Miroslav. On the complexity of some $\sigma$-ideals of $\sigma$-P-porous sets. Commentationes Mathematicae Universitatis Carolinae, Tome 44 (2003) no. 3, pp. 531-554. http://geodesic.mathdoc.fr/item/CMUC_2003_44_3_a11/