On the complexity of some $\sigma$-ideals of $\sigma$-P-porous sets

Zajíček, Luděk, and Zelený, Miroslav. "On the complexity of some $\sigma $-ideals of $\sigma $-P-porous sets." Commentationes Mathematicae Universitatis Carolinae 44.3 (2003): 531-554. <http://eudml.org/doc/249154>.

@article{Zajíček2003,
abstract = {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 $\Pi _\{\mathbf \{1\}\}^\{\mathbf \{1\}\}$-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. 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.},
author = {Zajíček, Luděk, Zelený, Miroslav},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {$\sigma $-porous sets; $\sigma $-ideal; coanalytic sets; Hausdorff metric; -porous sets; -ideal; coanalytic sets; Hausdorff metric},
language = {eng},
number = {3},
pages = {531-554},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On the complexity of some $\sigma $-ideals of $\sigma $-P-porous sets},
url = {http://eudml.org/doc/249154},
volume = {44},
year = {2003},
}

TY - JOUR
AU - Zajíček, Luděk
AU - Zelený, Miroslav
TI - On the complexity of some $\sigma $-ideals of $\sigma $-P-porous sets
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2003
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 44
IS - 3
SP - 531
EP - 554
AB - 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 $\Pi _{\mathbf {1}}^{\mathbf {1}}$-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. 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.
LA - eng
KW - $\sigma $-porous sets; $\sigma $-ideal; coanalytic sets; Hausdorff metric; -porous sets; -ideal; coanalytic sets; Hausdorff metric
UR - http://eudml.org/doc/249154
ER -