Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-16T17:03:34.176Z Has data issue: false hasContentIssue false

Descriptive set Theory of Families of Small Sets

Published online by Cambridge University Press:  15 January 2014

Étienne Matheron
Affiliation:
Université Bordeaux1, 351 Cours de la Libération, 33405 Talence Cedex, France, E-mail: [email protected]
Miroslav Zelený
Affiliation:
Charles University, Faculty of Mathematics and Physics, Department Of Mathematical Analysis, Sokolovská 83, 186 75 Prague, Czech Republic, E-mail: [email protected]

Abstract

This is a survey paper on the descriptive set theory of hereditary families of closed sets in Polish spaces. Most of the paper is devoted to ideals and σ-ideals of closed or compact sets.

Type
Articles
Copyright
Copyright © Association for Symbolic Logic 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[Ba] Balcerzak, M., A classification of σ-ideals in Polish groups, Demonstratio Mathematica, vol. 20 (1987), no. 1–2, pp. 7788.CrossRefGoogle Scholar
[BaRo] Balcerzak, M. and Rosłanowski, A., On Mycielski ideals, Proceedings of the American Mathematical Society, vol. 110 (1990), no. 1, pp. 243250.CrossRefGoogle Scholar
[Bary1] Bary, N. K., Sur l'funicité du développement trigonométrique, Fundamenta Mathematicae, vol. 9 (1927), pp. 62115.CrossRefGoogle Scholar
[Bary2] Bary, N. K., A treatise on trigonometric series, MacMillan, New York, 1964.Google Scholar
[BD] Balcerzak, M. and Darji, U. B., Some examples of true Fσδ sets, Colloquium Mathematicum, vol. 86 (2000), no. 2, pp. 203207.CrossRefGoogle Scholar
[BKL] Becker, H., Kahane, S., and Louveau, A., Some complete sets in harmonic analysis, Transactions of the American Mathematical Society, vol. 339 (1993), no. 1, pp. 323336.Google Scholar
[BKR] Bukovský, L., Kholshchevnikova, N. N., and Repický, M., Thin sets of harmonic analysis and infinite combinatorics, Real Analysis Exchange, vol. 20 (1994/1995), no. 2, pp. 454509.CrossRefGoogle Scholar
[BL] Benyamini, Y. and Lindenstrauss, J., Geometric nonlinear functional analysis, American Mathematical Society Colloquium Publications, vol. 1, American Mathematical Society, Providence, RI, 2000.Google Scholar
[BRS] Balcerzak, M., RosŁanowski, A., and Shelah, S., Ideals without ccc, The Journal of Symbolic Logic, vol. 63 (1998), no. 1, pp. 128148.CrossRefGoogle Scholar
[BS] Barua, R. and Srivatsa, V. V., Definable hereditary families in the projective hierarchy, Fundamenta Mathematicae, vol. 140 (1992), no. 2, pp. 183189.CrossRefGoogle Scholar
[Ca] Camerlo, R., Continua and their σ-ideals, Topology and its Applications, vol. 150 (2005), no. 1–3, pp. 118.CrossRefGoogle Scholar
[Cal1] Calbrix, J., Classes de Baire et espaces d'applications continues, Comptes Rendus de l'Académie des Sciences. Série I. Mathématique, vol. 301 (1985), no. 16, pp. 759762.Google Scholar
[Cal2] Calbrix, J., Filtres boréliens sur l'ensemble des entiers et espaces des applications continues, Revue Roumaine de Mathématiques Pures et Appliquées, vol. 33 (1988), no. 8, pp. 655661.Google Scholar
[Chr1] Christensen, J. P. R., On sets of Haar measure zero in abelian Polish groups, Israel Journal of Mathematics, vol. 13 (1972), pp. 255260.CrossRefGoogle Scholar
[Chr2] Christensen, J. P. R., Topology and Borel structure, North-Holland Mathematics Studies (Notas de Matemática), North-Holland, 1974.Google Scholar
[CM1] Cenzer, D. and Mauldin, R. D., On the Borel class of the derived set operator, Bulletin de la Société Mathématique de France, vol. 110 (1982), no. 4, pp. 357380.CrossRefGoogle Scholar
[CM2] Cenzer, D., On the Borel class of the derived set operator, II, Bulletin de la Société Mathématique de France, vol. 111 (1983), no. 4, pp. 367372.CrossRefGoogle Scholar
[D] Debs, G., Polar σ-ideals of compact sets, Transactions of the American Mathematical Society, vol. 347 (1995), no. 1, pp. 317338.Google Scholar
[Del] Dellacherie, C., Capacités et processus stochastiques, Springer-Verlag, Berlin, 1972, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 67.CrossRefGoogle Scholar
[DelMe] Dellacherie, C. and Meyer, P. A., Probabilités et potentiel. Chapitres IX à XI, Hermann, Paris, 1983, Actualités Scientifiques et Industrielles, 1410.Google Scholar
[Do1] Dougherty, R., Monotone but not positive subsets of the Cantor space, The Journal of Symbolic Logic, vol. 52 (1987), no. 3, pp. 817818.CrossRefGoogle Scholar
[Do2] Dougherty, R., Examples of nonshy sets, Fundamenta Mathematicae, vol. 144 (1994), no. 1, pp. 7388.CrossRefGoogle Scholar
[DSR] Debs, G. and Raymond, J. Saint, Ensembles boréliens d'unicité et d'unicité au sens large, Annales de l'Institut Fourier, vol. 37 (1987), no. 3, pp. 217239.CrossRefGoogle Scholar
[F1] Farah, I., Ideals induced by Tsirelson submeasures, Fundamenta Mathematicae, vol. 159 (1999), no. 3, pp. 243258.CrossRefGoogle Scholar
[F2] Farah, I., Analytic quotients: theory of liftings for quotients over analytic ideals on the integers, Memoirs of the American Mathematical Society, vol. 148 (2000).CrossRefGoogle Scholar
[FS] Farah, I. and Solecki, S., Two Fσδ ideals, Proceedings of the American Mathematical Society, vol. 131 (2003), no. 6, pp. 19711975.CrossRefGoogle Scholar
[FZ] Farah, I. and Zapletal, J., Four and more, Annals of Pure and Applied Logic, vol. 140 (2006), no. 1–3, pp. 339.CrossRefGoogle Scholar
[HKL] Harrington, L., Kechris, A. S., and Louveau, A., A Glimm–Effros dichotomy for Borel equivalence relations, Journal of the American Mathematical Society, vol. 3 (1990), no. 4, pp. 903928.CrossRefGoogle Scholar
[HP] Humke, P. D. and Preiss, D., Measures for which σ-porous sets are null, Journal of the London Mathematical Society. Second Series, vol. 32 (1985), no. 2, pp. 236244.CrossRefGoogle Scholar
[HSY] Hunt, B. R., Sauer, T., and Yorke, J. A., Prevalence: a translation invariant “almost every” on infinite-dimensional spaces, American Mathematical Society. Bulletin. New Series, vol. 27 (1992), no. 2, pp. 217238.CrossRefGoogle Scholar
[HT] Humke, P. D. and Thomson, B. S., A porosity characterization of symmetric perfect sets, Classical real analysis (Madison, Wis., 1982), Contemporary Mathematics, vol. 42, American Mathematical Society, Providence, RI, 1985, pp. 8185.CrossRefGoogle Scholar
[Hur] Hurewicz, W., Relativ perfekte Teile von Punktmengen und Mengen (A), Fundamenta Mathematicae, vol. 12 (1928), pp. 78109.CrossRefGoogle Scholar
[HZZ] Holický, P., Zajíček, L., and Zelený, M., A remark on a theorem of Solecki, Commentationes Mathematicae Universitatis Carolinae, vol. 46 (2005), no. 1, pp. 4354.Google Scholar
[J1] Jordan, F., Ideals of compact sets associated with Borel functions, Real Analysis Exchange, vol. 28 (2002/2003), no. 1, pp. 1531.CrossRefGoogle Scholar
[J2] Jordan, F., Collections of compact sets and functions having Gδ graph, preprint, 2007.Google Scholar
[K1] Kechris, A. S., Measure and category in effective descriptive set theory, Annals of Mathematical Logic, vol. 5 (1972/1973), pp. 337384.CrossRefGoogle Scholar
[K2] Kechris, A. S., The theory of countable analytical sets, Transactions of the American Mathematical Society, vol. 202 (1975), pp. 259297.CrossRefGoogle Scholar
[K3] Kechris, A. S., On a notion of smallness for subsets of the Baire space, Transactions of the American Mathematical Society, vol. 229 (1977), pp. 191207.CrossRefGoogle Scholar
[K4] Kechris, A. S., The descriptive set theory of σ-ideals of compact sets, Logic Colloquium '88 (Padova, 1988), Studies in Logic and the Foundations of Mathematics, vol. 127, North-Holland, Amsterdam, 1989, pp. 117138.Google Scholar
[K5] Kechris, A. S., Hereditary properties of the class of closed sets of uniqueness for trigonometric series, Israel Journal of Mathematics, vol. 73 (1991), no. 2, pp. 189198.CrossRefGoogle Scholar
[K6] Kechris, A. S., Classical descriptive set theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995.Google Scholar
[Kah] Kahane, J. P., Séries de Fourier absolument convergentes, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 50, Springer-Verlag, Berlin, 1970.CrossRefGoogle Scholar
[KahS1] Kahane, S., Ensembles de convergence absolue, ensembles de Dirichlet faibles et ↑-idéaux, Comptes Rendus de l'Académie des Sciences. Série I.Mathématique, vol. 310 (1990), no. 6, pp. 355357.Google Scholar
[KahS2] Kahane, S., Opérations de Hausdorff itérées et rèunions croissantes de compacts, Fundamenta Mathematicae, vol. 141 (1992), no. 2, pp. 169194.CrossRefGoogle Scholar
[KahS3] Kahane, S., Antistable classes of thin sets in harmonic analysis, Illinois Journal of Mathematics, vol. 37 (1993), no. 2, pp. 186223.CrossRefGoogle Scholar
[KahSa] Kahane, J. P. and Salem, R., Ensembles parfaits et séries trigonométriques, second ed., Hermann, Paris, 1994.Google Scholar
[Kau1] Kaufman, R., M-sets and distributions, Pseudofunctions and Helson sets, Astérisque, vol. 5, Société Mathématique de France, Paris, 1973, pp. 225230.Google Scholar
[Kau2] Kaufman, R., Fourier transforms and descriptive set theory, Mathematika, vol. 31 (1984), no. 2, pp. 336339.CrossRefGoogle Scholar
[Kau3] Kaufman, R., Absolutely convergent Fourier series and some classes of sets, Bulletin des Sciences Mathématiques. 2è Série, vol. 109 (1985), no. 4, pp. 363372.Google Scholar
[Kau4] Kaufman, R., M-sets and measures, Annals of Mathematics. Second Series, vol. 135 (1992), pp. 125130.CrossRefGoogle Scholar
[KL1] Kechris, A. S. and Louveau, A., Descriptive set theory and the structure of sets of uniqueness, London Mathematical Society Lecture Note Series, vol. 128, Cambridge University Press, Cambridge, 1987.CrossRefGoogle Scholar
[KL2] Kechris, A. S., Covering theorems for uniqueness and extended uniqueness sets, Colloquium Mathematicum, vol. 59 (1990), no. 1, pp. 6379.CrossRefGoogle Scholar
[KL3] Kechris, A. S., Descriptive set theory and harmonic analysis, The Journal of Symbolic Logic, vol. 57 (1992), no. 2, pp. 413441.CrossRefGoogle Scholar
[KLT] Kechris, A. S., Louveau, A., and Tardivel, V., The class of synthesizable pseudomeasures, Illinois Journal of Mathematics, vol. 35 (1991), no. 1, pp. 107146.CrossRefGoogle Scholar
[KLW] Kechris, A. S., Louveau, A., and Woodin, W. H., The structure of σ-ideals of compact sets, Transactions of the American Mathematical Society, vol. 301 (1987), no. 1, pp. 263288.Google Scholar
[KLy] Kechris, A. S. and Lyons, R., Ordinal rankings on measures anihilating thin sets, Transactions of the American Mathematical Society, vol. 310 (1988), no. 2, pp. 747758.CrossRefGoogle Scholar
[Ko1] Körner, T. W., Some results on Kronecker, Dirichlet and Helson sets, Annales de l'Institut Fourier, vol. 20 (1970), no. 2, pp. 219324.CrossRefGoogle Scholar
[Ko2] Körner, T. W., A pseudofunction on a Helson set I, Pseudofunctions and Helson sets, Astérisque, vol. 5, Société Mathématique de France, Paris, 1973, pp. 3224.Google Scholar
[Ko3] Körner, T. W., A pseudofunction on a Helson set II, Pseudofunctions and Helson sets, Astérisque, vol. 5, Société Mathématique de France, Paris, 1973, pp. 231239.Google Scholar
[Ko4] Körner, T. W., Some results on Kronecker, Dirichlet and Helson sets, II, Journal d'Analyse Mathématique, vol. 27 (1974), pp. 260388.CrossRefGoogle Scholar
[Ko5] Körner, T. W., A Helson set of uniqueness but not of synthesis, Colloquium Mathematicum, vol. 62 (1991), no. 1, pp. 6771.CrossRefGoogle Scholar
[Ko6] Körner, T. W., Variations on a theme of Debs and Saint Raymond, preprint, 2007.Google Scholar
[KS] Kechris, A. S. and Solecki, S., Approximation of analytic by Borel sets and definable countable chain conditions, Israel Journal of Mathematics, vol. 89 (1995), no. 1–3, pp. 343356.CrossRefGoogle Scholar
[L1] Louveau, A., Ensembles analytiques et boréliens dans les espaces produits, Astérisque, vol. 78, Société Mathématique de France, Paris, 1980.Google Scholar
[L2] Louveau, A., σ-idéaux engendrés par des ensembles fermés et théorèmes d'approximation, Transactions of the American Mathematical Society, vol. 257 (1980), no. 1, pp. 143169.Google Scholar
[L3] Louveau, A., Some results in the Wadge hierarchy of Borel sets, Cabal seminar 79–81, Lecture Notes in Mathematics, vol. 1019, Springer, Berlin, 1983, pp. 2855.CrossRefGoogle Scholar
[L4] Louveau, A., Sur la génération des fonctions boréliennes fortement affines sur un convexe compact métrisable, Annales de l'Institut Fourier, vol. 36 (1986), no. 2, pp. 5768.CrossRefGoogle Scholar
[Lin] Linton, T., The H-sets in the unit circle are properly Gδσ , Real Analysis Exchange, vol. 19 (1993/1994), no. 1, pp. 203211.CrossRefGoogle Scholar
[Loo] Loomis, L. H., The spectral characterization of a class of almost periodic functions, Annals of Mathematics. Second Series, vol. 72 (1960), no. 2, pp. 362368.CrossRefGoogle Scholar
[LP] Lindahl, L.-A. and Poulsen, F. (editors), Thin sets in harmonic analysis, Marcel Dekker Inc., New York, 1971.Google Scholar
[LPr] Lindenstrauss, J. and Preiss, D., On Fréchet differentiability of Lipschitz maps between Banach spaces, Annals of Mathematics. Second Series, vol. 157 (2003), no. 1, pp. 257288.CrossRefGoogle Scholar
[Ly1] Lyons, R., Fourier–Stieltjes coefficients and asymptotic distribution modulo 1, Annals of Mathematics. Second Series, vol. 122 (1985), no. 1, pp. 155170.CrossRefGoogle Scholar
[Ly2] Lyons, R., The size of some classes of thin sets, Studia Mathematica, vol. 86 (1987), no. 1, pp. 5978.CrossRefGoogle Scholar
[Ly3] Lyons, R., A new type of sets of uniqueness, Duke Mathematical Journal, vol. 57 (1988), no. 2, pp. 431458.CrossRefGoogle Scholar
[M1] Matheron, É., Sigma-ideáux polaires et ensembles d'unicité dans les groupes abéliens localement compacts, Annales de l'Institut Fourier, vol. 46 (1996), no. 2, pp. 493533.CrossRefGoogle Scholar
[M2] Matheron, É., How to recognize a true set, Fundamenta Mathematicae, vol. 158 (1998), no. 2, pp. 181194.CrossRefGoogle Scholar
[Mal] Malliavin, P., Ensembles de résolution spectrale, Proceedings of the International Congress of Mathematicians (Stockholm, 1962), Inst. Mittag-Leffler, Djursholm, 1963, pp. 368378.Google Scholar
[Marc] Marcone, A., Complexity of sets and binary relations in continuum theory: a survey, Set theory, Trends in Mathematics, Birkhäuser, Basel, 2006, pp. 121147.CrossRefGoogle Scholar
[MarKe] Martin, D. A. and Kechris, A. S., Infinite games and effective descriptive set theory, Analytic sets, Academic Press Inc., London, 1980, pp. 403470.Google Scholar
[Mau] Mauldin, R. D., σ-ideals and related Baire systems, Fundamenta Mathematicae, vol. 71 (1971), no. 2, pp. 171177.CrossRefGoogle Scholar
[MSZ] Matheron, É., Solecki, S., and Zelený, M., Trichotomies for ideals of compact sets, The Journal of Symbolic Logic, vol. 71 (2006), no. 2, pp. 586598.CrossRefGoogle Scholar
[Mva] Matoušková, E., Translating finite sets into convex sets, The Bulletin of the London Mathematical Society, vol. 33 (2001), no. 6, pp. 711714.CrossRefGoogle Scholar
[My] Mycielski, J., Independent sets in topological algebras, Fundamenta Mathematicae, vol. 55 (1964), pp. 139147.CrossRefGoogle Scholar
[MZ] Matheron, É. and Zelený, M., Rudin-like sets and hereditary families of compact sets, Fundamenta Mathematicae, vol. 185 (2005), no. 2, pp. 97116.CrossRefGoogle Scholar
[OY] Ott, W. and Yorke, J. A., Prevalence, American Mathematical Society. Bulletin. New Series, vol. 42 (2005), no. 3, pp. 263290.CrossRefGoogle Scholar
[PS] Pyateckiĭ-Šapiro, I. I., Supplement to the work “On the problem of uniqueness of expansion of a function in a trigonometric series.”, Moskov. Gos. Univ. Uč. Zap. Mat., vol. 165 (1954), no. 7, pp. 7997.Google Scholar
[Ptr1] Petruska, G., On Borel sets with small cover: a problem of M. Laczkovich, Real Analysis Exchange, vol. 18 (1992/1993), no. 2, pp. 330338.CrossRefGoogle Scholar
[Ptr2] Petruska, G., Errata to: “On Borel sets with small cover: a problem of M. Laczkovich”, Real Analysis Exchange, vol. 19 (1993/1994), no. 1, p. 58.CrossRefGoogle Scholar
[Ru] Rudin, W., Fourier–Stieltjes transforms of measures on independent sets, Bulletin of the American Mathematical Society, vol. 66 (1960), pp. 199202.CrossRefGoogle Scholar
[RZ] Recław, I. and Zakrzewski, P., Strong Fubini properties of ideals, Fundamenta Mathematicae, vol. 159 (1999), no. 2, pp. 135152.CrossRefGoogle Scholar
[S1] Solecki, S., Covering analytic sets by families of closed sets, The Journal of Symbolic Logic, vol. 59 (1994), no. 3, pp. 10221031.CrossRefGoogle Scholar
[S2] Solecki, S., On Haar null sets, Fundamenta Mathematicae, vol. 149 (1995), no. 3, pp. 205210.CrossRefGoogle Scholar
[S3] Solecki, S., Analytic ideals, this Bulletin, vol. 2 (1996), no. 3, pp. 339348.Google Scholar
[S4] Solecki, S., Analytic ideals and their applications, Annals of Pure and Applied Logic, vol. 99 (1999), no. 1–3, pp. 5172.CrossRefGoogle Scholar
[S5] Solecki, S., Haar null and non-dominating sets, Fundamenta Mathematicae, vol. 170 (2001), no. 1–2, pp. 197217.CrossRefGoogle Scholar
[S6] Solecki, S., Size of subsets of groups and Haar null sets, Geometric and Functional Analysis, vol. 15 (2005), no. 1, pp. 246273.CrossRefGoogle Scholar
[S7] Solecki, S., Amenability, free subgroups, and Haar null sets in non-locally compact groups, Proceedings of the London Mathematical Society. Third Series, vol. 93 (2006), no. 3, pp. 693722.CrossRefGoogle Scholar
[S8] Solecki, S., Cofinal Gδ subsets of analytic ideals of compact sets, preprint, 2007.Google Scholar
[S9] Solecki, S., Ideals of compact sets, preprint, 2007.Google Scholar
[S1] Šleich, P., Sets of type H(s) are σ-bilaterally porous, unpublished.Google Scholar
[SR1] Raymond, J. Saint, Caractérisation d'espaces polonais. D'après des travaux récents de J. P. R. Christensen et D. Preiss, Séminaire Choquet, initiation à l'analyse, 11è–12è années (1971–1973), exposé no. 5, Secrétariat Mathématique, Paris, 1973, p. 10.Google Scholar
[SR2] Raymond, J. Saint, Approximation des sous-ensembles analytiques par l'intérieur, Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences. Séries A et B, vol. 281 (1975), no. 2–3, pp. Aii, A85A87.Google Scholar
[SR3] Raymond, J. Saint, La structure borélienne d'Effros est-elle standard?, Fundamenta Mathematicae, vol. 100 (1978), no. 3, pp. 201210.CrossRefGoogle Scholar
[SR4] Raymond, J. Saint, Quasi-bounded trees and analytic inductions, Fundamenta Mathematicae, vol. 191 (2006), no. 2, pp. 175185.CrossRefGoogle Scholar
[ST] Solecki, S. and Todorcevic, S., Cofinal types of topological directed orders, Annales de l'Institut Fourier, vol. 54 (2004), no. 6, pp. 18771911.CrossRefGoogle Scholar
[T] Tardivel, V., Fermés d'unicité dans les groupes abéliens localement compacts, Studia Mathematica, vol. 91 (1988), no. 1, pp. 115.CrossRefGoogle Scholar
[U1] Uzcátegui, C. E., The covering property for σ-ideals of compact sets, Fundamenta Mathematicae, vol. 141 (1992), no. 2, pp. 119146.CrossRefGoogle Scholar
[U2] Uzcátegui, C. E., Smooth sets for a Borel equivalence relation, Transactions of the American Mathematical Society, vol. 347 (1995), no. 6, pp. 20252039.CrossRefGoogle Scholar
[U3] Uzcátegui, C. E., The σ-ideal of closed smooth sets does not have the covering property, Fundamenta Mathematicae, vol. 150 (1996), no. 3, pp. 227236.CrossRefGoogle Scholar
[V] Veličković, B., A note on Tsirelson type ideals, Fundamenta Mathematicae, vol. 159 (1999), no. 3, pp. 259268.CrossRefGoogle Scholar
[vE] van Engelen, F., On Borel ideals, Annals of Pure and Applied Logic, vol. 70 (1994), no. 2, pp. 177203.CrossRefGoogle Scholar
[V1] Vlasák, V., Compact sets of continuity for Borel functions, in preparation.Google Scholar
[Za1] Zajíček, L., Porosity and σ-porosity, Real Analysis Exchange, vol. 13 (1987/1988), no. 2, pp. 314350.CrossRefGoogle Scholar
[Za2] Zajíček, L., Errata: “Porosity and σ-porosity”, Real Analysis Exchange, vol. 14 (1988/1989), no. 1, p. 5.Google Scholar
[Za3] Zajíček, L., An unpublished result of P. Šleich: sets of type H(s) are σ-bilaterally porous, Real Analysis Exchange, vol. 27 (2001/2002), no. 1, pp. 363372.CrossRefGoogle Scholar
[Za4] Zajíček, L., On σ-porous sets in abstract spaces, Abstract and Applied Analysis, (2005), no. 5, pp. 509534.Google Scholar
[Zaf1] Zafrany, S., Borel ideals vs. Borel sets of countable relations and trees, Annals of Pure and Applied Logic, vol. 43 (1989), no. 2, pp. 161195.CrossRefGoogle Scholar
[Zaf2] Zafrany, S., On analytic filters and prefilters, The Journal of Symbolic Logic, vol. 55 (1990), no. 1, pp. 315322.CrossRefGoogle Scholar
[Zap] Zapletal, J., Forcing idealized, Cambridge Tracts in Mathematics, vol. 174, Cambridge University Press, to appear.Google Scholar
[ZaZe1] Zajíček, L. and Zelený, M., On the complexity of some σ-ideals of σ-P-porous sets, Commentationes Mathematicae Universitatis Carolinae, vol. 44 (2003), no. 3, pp. 531554.Google Scholar
[ZaZe2] Zajíček, L., Inscribing closed non-σ-lower porous sets into Suslin non-σ-lower porous sets, Abstract and Applied Analysis, (2005), no. 3, pp. 221227.Google Scholar
[Ze1] Zelený, M., The Banach–Mazur game and σ-porosity, Fundamenta Mathematicae, vol. 150 (1996), no. 3, pp. 197210.CrossRefGoogle Scholar
[Ze2] Zelený, M., Calibrated thin σ-ideals are Gδ , Proceedings of the American Mathematical Society, vol. 125 (1997), no. 10, pp. 30273032.CrossRefGoogle Scholar
[ZeZa] Zelený, M. and Zajíček, L., Inscribing compact non-σ-porous sets into analytic non-σ-porous sets, Fundamenta Mathematicae, vol. 185 (2005), no. 1, pp. 1939.CrossRefGoogle Scholar
[ZP] Zelený, M. and Pelant, J., The structure of the σ-ideal of σ-porous sets, Commentationes Mathematicae Universitatis Carolinae, vol. 45 (2004), no. 1, pp. 3772.Google Scholar