# The structure of the $\sigma $-ideal of $\sigma $-porous sets

Commentationes Mathematicae Universitatis Carolinae (2004)

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

## Access Full Article

top## Abstract

top## How to cite

topZelený, 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 -

## References

top- Bari N., Trigonometric Series, Moscow, 1961. Zbl0154.06103MR0126115
- Becker H., Kahane S., Louveau A., Some complete $\AA $ sets in harmonic analysis, Trans. Amer. Math. Soc. 339 (1993), 1 323-336. (1993) MR1129434
- Bukovský L., Kholshchevnikova N.N., Repický M., Thin sets of harmonic analysis and infinite combinatorics, Real Anal. Exchange 20 (1994-95), 2 454-509. (1994-95) MR1348075
- Debs G., Private communication, .
- Dolzhenko E.P., Boundary properties of arbitrary functions, Izv. Akad. Nauk SSSR Ser. Mat. 31 (1967), 3-14 (in Russian). (1967) MR0217297
- Debs G., Saint-Raymond J., Ensembles boréliens d'unicité au sens large, Ann. Inst. Fourier (Grenoble) 37 (1987), 3 217-239. (1987) MR0916281
- Kaufman R., Fourier transforms and descriptive set theory, Mathematika 31 (1984), 2 336-339. (1984) Zbl0604.42009MR0804207
- Kechris A.S., Classical Descriptive Set Theory, Springer-Verlag, New York, 1995. Zbl0819.04002MR1321597
- Kechris A.S., Louveau A., Descriptive Set Theory and the Structure of Sets of Uniqueness, London Math. Soc. Lecture Notes Series 128, Cambridge University Press, Cambridge, 1989. Zbl0677.42009MR0953784
- Kechris A.S., Louveau A., Woodin W.H., The structure of $\sigma $-ideals of compact sets, Trans. Amer. Math. Soc. 301 (1987), 1 263-288. (1987) Zbl0633.03043MR0879573
- Laczkovich M., Analytic subgroups of the reals, Proc. Amer. Math. Soc. 126 (1998), 6 1783-1790. (1998) Zbl0896.04002MR1443837
- Loomis L., The spectral characterization of a class of almost periodic functions, Ann. of Math. 72 (1960), 2 362-368. (1960) Zbl0094.05801MR0120502
- Lindahl L.-A., Poulsen F., Thin Sets in Harmonic Analysis, Marcel Dekker, New York, 1971. Zbl0226.43006MR0393993
- Piatetski-Shapiro I.I., On the problem of uniqueness expansion of a function in a trigonometric series, Moscov. Gos. Univ. Uchen. Zap., vol. 155, Mat. 5 (1952), 54-72. MR0080201
- Rogers C.A. et al., Analytic Sets, Academic Press, London, 1980. Zbl0589.54047MR0608794
- Reclaw I., A note on the $\sigma $-ideal of $\sigma $-porous sets, Real Anal. Exchange 12 (1986-87), 2 455-457. (1986-87) Zbl0656.26001MR0888722
- Solecki S., Covering analytic sets by families of closed sets, J. Symbolic Logic 59 (1994), 3 1022-1031. (1994) Zbl0808.03031MR1295987
- Šleich P., Sets of type ${H}^{\left(s\right)}$ are $\sigma $-bilaterally porous, preprint (unpublished).
- Zajíček L., Sets of $\sigma $-porosity and $\sigma $-porosity $\left(q\right)$, Časopis Pěst. Mat. 101 (1976), 4 350-359. (1976) MR0457731
- Zajíček L., Porosity and $\sigma $-porosity, Real Anal. Exchange 13 (1987-88), 2 314-350. (1987-88) MR0943561
- Zajíček L., Small non-sigma-porous sets in topologically complete metric spaces, Colloq. Math. 77 (1998), 2 293-304. (1998) MR1628994
- Zajíček L., Smallness of sets of nondifferentiability of convex functions in non-separable Banach spaces, Czechoslovak Math. J. 41 (116) (1991), 288-296. (1991) MR1105445
- Zajíček L., An unpublished result of P. Sleich: sets of type ${H}^{\left(s\right)}$ are $\sigma $-bilaterally porous, Real Anal. Exchange 27 (2002), 1 363-372. (2002) MR1887868
- Zelený M., Calibrated thin ${\Pi}_{\mathbf{1}}^{\mathbf{1}}$$\sigma $-ideals are ${G}_{\delta}$, Proc. Amer. Math. Soc. 125 (1997), 10 3027-3032. (1997) MR1415378
- Zelený M., On singular boundary points of complex functions, Mathematika 45 (1998), 1 119-133. (1998) MR1644354

## Citations in EuDML Documents

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

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.