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Descriptive set Theory of Families of Small Sets

Published online by Cambridge University Press:  15 January 2014

Étienne Matheron  and
Miroslav Zelený
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]
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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.

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Articles
Information
Bulletin of Symbolic Logic , Volume 13 , Issue 4 , December 2007 , pp. 482 - 537
Copyright
Copyright © Association for Symbolic Logic 2007

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References

REFERENCES

[Ba] Balcerzak, M., A classification of σ-ideals in Polish groups, Demonstratio Mathematica, vol. 20 (1987), no. 1–2, pp. 77–88.CrossRefGoogle Scholar
[BaRo] Balcerzak, M. and Rosłanowski, A., On Mycielski ideals, Proceedings of the American Mathematical Society, vol. 110 (1990), no. 1, pp. 243–250.CrossRefGoogle Scholar
[Bary1] Bary, N. K., Sur l'funicité du développement trigonométrique, Fundamenta Mathematicae, vol. 9 (1927), pp. 62–115.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. 203–207.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. 323–336.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. 454–509.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. 128–148.CrossRefGoogle Scholar
[BS] Barua, R. and Srivatsa, V. V., Definable hereditary families in the projective hierarchy, Fundamenta Mathematicae, vol. 140 (1992), no. 2, pp. 183–189.CrossRefGoogle Scholar
[Ca] Camerlo, R., Continua and their σ-ideals, Topology and its Applications, vol. 150 (2005), no. 1–3, pp. 1–18.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. 759–762.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. 655–661.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. 255–260.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. 357–380.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. 367–372.CrossRefGoogle Scholar
[D] Debs, G., Polar σ-ideals of compact sets, Transactions of the American Mathematical Society, vol. 347 (1995), no. 1, pp. 317–338.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. 817–818.CrossRefGoogle Scholar
[Do2] Dougherty, R., Examples of nonshy sets, Fundamenta Mathematicae, vol. 144 (1994), no. 1, pp. 73–88.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. 217–239.CrossRefGoogle Scholar
[F1] Farah, I., Ideals induced by Tsirelson submeasures, Fundamenta Mathematicae, vol. 159 (1999), no. 3, pp. 243–258.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. 1971–1975.CrossRefGoogle Scholar
[FZ] Farah, I. and Zapletal, J., Four and more, Annals of Pure and Applied Logic, vol. 140 (2006), no. 1–3, pp. 3–39.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. 903–928.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. 236–244.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. 217–238.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. 81–85.CrossRefGoogle Scholar
[Hur] Hurewicz, W., Relativ perfekte Teile von Punktmengen und Mengen (A), Fundamenta Mathematicae, vol. 12 (1928), pp. 78–109.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. 43–54.Google Scholar
[J1] Jordan, F., Ideals of compact sets associated with Borel functions, Real Analysis Exchange, vol. 28 (2002/2003), no. 1, pp. 15–31.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. 337–384.CrossRefGoogle Scholar
[K2] Kechris, A. S., The theory of countable analytical sets, Transactions of the American Mathematical Society, vol. 202 (1975), pp. 259–297.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. 191–207.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. 117–138.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. 189–198.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. 355–357.Google Scholar
[KahS2] Kahane, S., Opérations de Hausdorff itérées et rèunions croissantes de compacts, Fundamenta Mathematicae, vol. 141 (1992), no. 2, pp. 169–194.CrossRefGoogle Scholar
[KahS3] Kahane, S., Antistable classes of thin sets in harmonic analysis, Illinois Journal of Mathematics, vol. 37 (1993), no. 2, pp. 186–223.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. 225–230.Google Scholar
[Kau2] Kaufman, R., Fourier transforms and descriptive set theory, Mathematika, vol. 31 (1984), no. 2, pp. 336–339.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. 363–372.Google Scholar
[Kau4] Kaufman, R., M-sets and measures, Annals of Mathematics. Second Series, vol. 135 (1992), pp. 125–130.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. 63–79.CrossRefGoogle Scholar
[KL3] Kechris, A. S., Descriptive set theory and harmonic analysis, The Journal of Symbolic Logic, vol. 57 (1992), no. 2, pp. 413–441.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. 107–146.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. 263–288.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. 747–758.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. 219–324.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. 3–224.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. 231–239.Google Scholar
[Ko4] Körner, T. W., Some results on Kronecker, Dirichlet and Helson sets, II, Journal d'Analyse Mathématique, vol. 27 (1974), pp. 260–388.CrossRefGoogle Scholar
[Ko5] Körner, T. W., A Helson set of uniqueness but not of synthesis, Colloquium Mathematicum, vol. 62 (1991), no. 1, pp. 67–71.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. 343–356.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. 143–169.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. 28–55.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. 57–68.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. 203–211.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. 362–368.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. 257–288.CrossRefGoogle Scholar
[Ly1] Lyons, R., Fourier–Stieltjes coefficients and asymptotic distribution modulo 1, Annals of Mathematics. Second Series, vol. 122 (1985), no. 1, pp. 155–170.CrossRefGoogle Scholar
[Ly2] Lyons, R., The size of some classes of thin sets, Studia Mathematica, vol. 86 (1987), no. 1, pp. 59–78.CrossRefGoogle Scholar
[Ly3] Lyons, R., A new type of sets of uniqueness, Duke Mathematical Journal, vol. 57 (1988), no. 2, pp. 431–458.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. 493–533.CrossRefGoogle Scholar
[M2] Matheron, É., How to recognize a true set, Fundamenta Mathematicae, vol. 158 (1998), no. 2, pp. 181–194.CrossRefGoogle Scholar
[Mal] Malliavin, P., Ensembles de résolution spectrale, Proceedings of the International Congress of Mathematicians (Stockholm, 1962), Inst. Mittag-Leffler, Djursholm, 1963, pp. 368–378.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. 121–147.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. 403–470.Google Scholar
[Mau] Mauldin, R. D., σ-ideals and related Baire systems, Fundamenta Mathematicae, vol. 71 (1971), no. 2, pp. 171–177.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. 586–598.CrossRefGoogle Scholar
[Mva] Matoušková, E., Translating finite sets into convex sets, The Bulletin of the London Mathematical Society, vol. 33 (2001), no. 6, pp. 711–714.CrossRefGoogle Scholar
[My] Mycielski, J., Independent sets in topological algebras, Fundamenta Mathematicae, vol. 55 (1964), pp. 139–147.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. 97–116.CrossRefGoogle Scholar
[OY] Ott, W. and Yorke, J. A., Prevalence, American Mathematical Society. Bulletin. New Series, vol. 42 (2005), no. 3, pp. 263–290.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. 79–97.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. 330–338.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. 199–202.CrossRefGoogle Scholar
[RZ] Recław, I. and Zakrzewski, P., Strong Fubini properties of ideals, Fundamenta Mathematicae, vol. 159 (1999), no. 2, pp. 135–152.CrossRefGoogle Scholar
[S1] Solecki, S., Covering analytic sets by families of closed sets, The Journal of Symbolic Logic, vol. 59 (1994), no. 3, pp. 1022–1031.CrossRefGoogle Scholar
[S2] Solecki, S., On Haar null sets, Fundamenta Mathematicae, vol. 149 (1995), no. 3, pp. 205–210.CrossRefGoogle Scholar
[S3] Solecki, S., Analytic ideals, this Bulletin, vol. 2 (1996), no. 3, pp. 339–348.Google Scholar
[S4] Solecki, S., Analytic ideals and their applications, Annals of Pure and Applied Logic, vol. 99 (1999), no. 1–3, pp. 51–72.CrossRefGoogle Scholar
[S5] Solecki, S., Haar null and non-dominating sets, Fundamenta Mathematicae, vol. 170 (2001), no. 1–2, pp. 197–217.CrossRefGoogle Scholar
[S6] Solecki, S., Size of subsets of groups and Haar null sets, Geometric and Functional Analysis, vol. 15 (2005), no. 1, pp. 246–273.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. 693–722.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, A85–A87.Google Scholar
[SR3] Raymond, J. Saint, La structure borélienne d'Effros est-elle standard?, Fundamenta Mathematicae, vol. 100 (1978), no. 3, pp. 201–210.CrossRefGoogle Scholar
[SR4] Raymond, J. Saint, Quasi-bounded trees and analytic inductions, Fundamenta Mathematicae, vol. 191 (2006), no. 2, pp. 175–185.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. 1877–1911.CrossRefGoogle Scholar
[T] Tardivel, V., Fermés d'unicité dans les groupes abéliens localement compacts, Studia Mathematica, vol. 91 (1988), no. 1, pp. 1–15.CrossRefGoogle Scholar
[U1] Uzcátegui, C. E., The covering property for σ-ideals of compact sets, Fundamenta Mathematicae, vol. 141 (1992), no. 2, pp. 119–146.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. 2025–2039.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. 227–236.CrossRefGoogle Scholar
[V] Veličković, B., A note on Tsirelson type ideals, Fundamenta Mathematicae, vol. 159 (1999), no. 3, pp. 259–268.CrossRefGoogle Scholar
[vE] van Engelen, F., On Borel ideals, Annals of Pure and Applied Logic, vol. 70 (1994), no. 2, pp. 177–203.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. 314–350.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. 363–372.CrossRefGoogle Scholar
[Za4] Zajíček, L., On σ-porous sets in abstract spaces, Abstract and Applied Analysis, (2005), no. 5, pp. 509–534.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. 161–195.CrossRefGoogle Scholar
[Zaf2] Zafrany, S., On analytic filters and prefilters, The Journal of Symbolic Logic, vol. 55 (1990), no. 1, pp. 315–322.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. 531–554.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. 221–227.Google Scholar
[Ze1] Zelený, M., The Banach–Mazur game and σ-porosity, Fundamenta Mathematicae, vol. 150 (1996), no. 3, pp. 197–210.CrossRefGoogle Scholar
[Ze2] Zelený, M., Calibrated thin σ-ideals are Gδ , Proceedings of the American Mathematical Society, vol. 125 (1997), no. 10, pp. 3027–3032.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. 19–39.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. 37–72.Google Scholar