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

Luděk Zajíček; Miroslav Zelený

Commentationes Mathematicae Universitatis Carolinae (2003)

- Volume: 44, Issue: 3, page 531-554
- ISSN: 0010-2628

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topZajíček, 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 -

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