The structure of the σ -ideal of σ -porous sets

Miroslav Zelený; Jan Pelant

Commentationes Mathematicae Universitatis Carolinae (2004)

  • Volume: 45, Issue: 1, page 37-72
  • ISSN: 0010-2628

Abstract

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We show a general method of construction of non- σ -porous sets in complete metric spaces. This method enables us to answer several open questions. We prove that each non- σ -porous Suslin subset of a topologically complete metric space contains a non- σ -porous closed subset. We show also a sufficient condition, which gives that a certain system of compact sets contains a non- σ -porous element. Namely, if we denote the space of all compact subsets of a compact metric space E with the Vietoris topology by 𝒦 ( E ) , then it is shown that each analytic subset of 𝒦 ( E ) containing all countable compact subsets of E contains necessarily an element, which is a non- σ -porous subset of E . We show several applications of this result to problems from real and harmonic analysis (e.g. the existence of a closed non- σ -porous set of uniqueness for trigonometric series). Finally we investigate also descriptive properties of the σ -ideal of compact σ -porous sets.

How to cite

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Zelený, Miroslav, and Pelant, Jan. "The structure of the $\sigma $-ideal of $\sigma $-porous sets." Commentationes Mathematicae Universitatis Carolinae 45.1 (2004): 37-72. <http://eudml.org/doc/249325>.

@article{Zelený2004,
abstract = {We show a general method of construction of non-$\sigma $-porous sets in complete metric spaces. This method enables us to answer several open questions. We prove that each non-$\sigma $-porous Suslin subset of a topologically complete metric space contains a non-$\sigma $-porous closed subset. We show also a sufficient condition, which gives that a certain system of compact sets contains a non-$\sigma $-porous element. Namely, if we denote the space of all compact subsets of a compact metric space $E$ with the Vietoris topology by $\mathcal \{K\}(E)$, then it is shown that each analytic subset of $\mathcal \{K\}(E)$ containing all countable compact subsets of $E$ contains necessarily an element, which is a non-$\sigma $-porous subset of $E$. We show several applications of this result to problems from real and harmonic analysis (e.g. the existence of a closed non-$\sigma $-porous set of uniqueness for trigonometric series). Finally we investigate also descriptive properties of the $\sigma $-ideal of compact $\sigma $-porous sets.},
author = {Zelený, Miroslav, Pelant, Jan},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {$\sigma $-porosity; descriptive set theory; $\sigma $-ideal; trigonometric series; sets of uniqueness; -porosity; descriptive set theory; trigonometric series; sets of uniqueness},
language = {eng},
number = {1},
pages = {37-72},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {The structure of the $\sigma $-ideal of $\sigma $-porous sets},
url = {http://eudml.org/doc/249325},
volume = {45},
year = {2004},
}

TY - JOUR
AU - Zelený, Miroslav
AU - Pelant, Jan
TI - The structure of the $\sigma $-ideal of $\sigma $-porous sets
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2004
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 45
IS - 1
SP - 37
EP - 72
AB - We show a general method of construction of non-$\sigma $-porous sets in complete metric spaces. This method enables us to answer several open questions. We prove that each non-$\sigma $-porous Suslin subset of a topologically complete metric space contains a non-$\sigma $-porous closed subset. We show also a sufficient condition, which gives that a certain system of compact sets contains a non-$\sigma $-porous element. Namely, if we denote the space of all compact subsets of a compact metric space $E$ with the Vietoris topology by $\mathcal {K}(E)$, then it is shown that each analytic subset of $\mathcal {K}(E)$ containing all countable compact subsets of $E$ contains necessarily an element, which is a non-$\sigma $-porous subset of $E$. We show several applications of this result to problems from real and harmonic analysis (e.g. the existence of a closed non-$\sigma $-porous set of uniqueness for trigonometric series). Finally we investigate also descriptive properties of the $\sigma $-ideal of compact $\sigma $-porous sets.
LA - eng
KW - $\sigma $-porosity; descriptive set theory; $\sigma $-ideal; trigonometric series; sets of uniqueness; -porosity; descriptive set theory; trigonometric series; sets of uniqueness
UR - http://eudml.org/doc/249325
ER -

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Citations in EuDML Documents

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  1. Szymon Gła̧b, Descriptive set-theoretical properties of an abstract density operator
  2. Michael Dymond, On the structure of universal differentiability sets
  3. Martin Rmoutil, Products of non- σ -lower porous sets
  4. Viktoriia Bilet, Oleksiy Dovgoshey, Jürgen Prestin, Two ideals connected with strong right upper porosity at a point
  5. Marek Cúth, Martin Rmoutil, σ -porosity is separably determined
  6. Bohuslav Balcar, Vladimír Müller, Jaroslav Nešetřil, Petr Simon, Jan Pelant (18.2.1950–11.4.2005)
  7. Bohuslav Balcar, Vladimír Müller, Jaroslav Nešetřil, Petr Simon, Jan Pelant (18.2.1950–11.4.2005)

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