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

top
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

top

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 -

References

top
  1. Bari N., Trigonometric Series, Moscow, 1961. Zbl0154.06103MR0126115
  2. Becker H., Kahane S., Louveau A., Some complete Å sets in harmonic analysis, Trans. Amer. Math. Soc. 339 (1993), 1 323-336. (1993) MR1129434
  3. 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
  4. Debs G., Private communication, . 
  5. Dolzhenko E.P., Boundary properties of arbitrary functions, Izv. Akad. Nauk SSSR Ser. Mat. 31 (1967), 3-14 (in Russian). (1967) MR0217297
  6. Debs G., Saint-Raymond J., Ensembles boréliens d'unicité au sens large, Ann. Inst. Fourier (Grenoble) 37 (1987), 3 217-239. (1987) MR0916281
  7. Kaufman R., Fourier transforms and descriptive set theory, Mathematika 31 (1984), 2 336-339. (1984) Zbl0604.42009MR0804207
  8. Kechris A.S., Classical Descriptive Set Theory, Springer-Verlag, New York, 1995. Zbl0819.04002MR1321597
  9. 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
  10. Kechris A.S., Louveau A., Woodin W.H., The structure of σ -ideals of compact sets, Trans. Amer. Math. Soc. 301 (1987), 1 263-288. (1987) Zbl0633.03043MR0879573
  11. Laczkovich M., Analytic subgroups of the reals, Proc. Amer. Math. Soc. 126 (1998), 6 1783-1790. (1998) Zbl0896.04002MR1443837
  12. Loomis L., The spectral characterization of a class of almost periodic functions, Ann. of Math. 72 (1960), 2 362-368. (1960) Zbl0094.05801MR0120502
  13. Lindahl L.-A., Poulsen F., Thin Sets in Harmonic Analysis, Marcel Dekker, New York, 1971. Zbl0226.43006MR0393993
  14. 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
  15. Rogers C.A. et al., Analytic Sets, Academic Press, London, 1980. Zbl0589.54047MR0608794
  16. Reclaw I., A note on the σ -ideal of σ -porous sets, Real Anal. Exchange 12 (1986-87), 2 455-457. (1986-87) Zbl0656.26001MR0888722
  17. Solecki S., Covering analytic sets by families of closed sets, J. Symbolic Logic 59 (1994), 3 1022-1031. (1994) Zbl0808.03031MR1295987
  18. Šleich P., Sets of type H ( s ) are σ -bilaterally porous, preprint (unpublished). 
  19. Zajíček L., Sets of σ -porosity and σ -porosity ( q ) , Časopis Pěst. Mat. 101 (1976), 4 350-359. (1976) MR0457731
  20. Zajíček L., Porosity and σ -porosity, Real Anal. Exchange 13 (1987-88), 2 314-350. (1987-88) MR0943561
  21. Zajíček L., Small non-sigma-porous sets in topologically complete metric spaces, Colloq. Math. 77 (1998), 2 293-304. (1998) MR1628994
  22. 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
  23. Zajíček L., An unpublished result of P. Sleich: sets of type H ( s ) are σ -bilaterally porous, Real Anal. Exchange 27 (2002), 1 363-372. (2002) MR1887868
  24. Zelený M., Calibrated thin Π 1 1 σ -ideals are G δ , Proc. Amer. Math. Soc. 125 (1997), 10 3027-3032. (1997) MR1415378
  25. Zelený M., On singular boundary points of complex functions, Mathematika 45 (1998), 1 119-133. (1998) MR1644354

Citations in EuDML Documents

top
  1. Szymon Gła̧b, Descriptive set-theoretical properties of an abstract density operator
  2. Michael Dymond, On the structure of universal differentiability sets
  3. Viktoriia Bilet, Oleksiy Dovgoshey, Jürgen Prestin, Two ideals connected with strong right upper porosity at a point
  4. Martin Rmoutil, Products of non- σ -lower porous sets
  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)

NotesEmbed ?

top

You must be logged in to post comments.

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

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.