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Descriptive classes of sets in nonseparable spaces

Clases descriptivas de conjuntos en espacios no separables

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Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas Aims and scope Submit manuscript

Abstract

Several attempts to transfer methods of the classical descriptive set theory to study measurability in nonseparable metrizable or more general topological spaces are presented. In particular, the theories of K-analytic spaces and of absolute Souslin metrizable spaces give quite interesting results. The spaces which are results of the Souslin operation applied to the families of Borel sets (Čech-analytic spaces) or of resolvable sets in every Tychonoff space allow further generalizations. Many nonseparable Banach spaces with their weak topology may serve as examples of such spaces. We recall a number of results and we point out open questions which seem to make the extension of the classical descriptive set theory and its applications to functional analysis not yet sufficiently complete.

Resumen

Se presentan varios enfoques para transferir métodos de la teoría clásica descriptiva de conjuntos al estudio de medibilidad en espacios metrizables no separables e incluso en espacios más generales. En particular, las teorías de espacios K-analíticos y de espacios absolutamente Souslin (en espacios metrizables) dan resultados muy interesantes. Otras generalizaciones las proporcionan los espacios obtenidos al aplicar la operación de Souslin a familias de conjuntos de Borel (espacios Čech-analíticos), así como los conjuntos resolubles en cada espacio de Tychonoff. Muchos espacios de Banach no separables provistos con la topología débil suministran ejemplos de estos espacios. Se dan bastantes resultados y se señalan cuestiones abiertas que muestran que la extensión al análisis funcional y a sus aplicaciones de la teoría clásica descriptiva de conjuntos no está todavía suficientemente completa.

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References

  1. Argyros, S. A.; Arvanitakis, A. D. and Mercourakis, S. K., (2008). Talagrand’s K σ δ problem, Topology Appl., 155, 1737–1755. DOI: 10.1016/j.topol.2008.05.014

    Article  MATH  MathSciNet  Google Scholar 

  2. Dellacherie, C., (1980). Un cours sur les ensembles analytiques, in Analytic sets, C. A. Rogers, ed., Academic Press, London, 183–316.

    Google Scholar 

  3. Edgar, G. A., (1977). Measurability in a Banach space, Indiana Univ. Math. J., 26, 663–677.

    Article  MATH  MathSciNet  Google Scholar 

  4. Engelking, R., (1989). General Topology, Heldermann Verlag, Berlin.

    MATH  Google Scholar 

  5. Fleissner, W. G., (1979). An axiom for nonseparable Borel theory, Trans. Amer. Math. Soc., 251, 309–328. DOI: 10.2307/1998696

    Article  MATH  MathSciNet  Google Scholar 

  6. Fremlin, D. H., (1980). Čech-analytic spaces, unpublished note.

  7. Fremlin, D. H.; Hansell, R. W. and Junnila, H. J. K., (1983). Borel functions of bounded class, Trans. Amer. Math. Soc., 277, 2, 835–849. DOI: 10.2307/1999240

    Article  MATH  MathSciNet  Google Scholar 

  8. Frolík, Z., (1970). A survey of separable descriptive theory of sets and spaces, Czechoslovak Math. J., 20, 406–467.

    MathSciNet  Google Scholar 

  9. Frolík, Z., (1970). A measurable map with analytic domain and metrizable range is quotient, Bull. Amer. Math. Soc., 76, 1112–1117. DOI: 10.1090/S0002-9904-1970-12584-8

    Article  MATH  MathSciNet  Google Scholar 

  10. Frolík, Z., (1984). Distinguished subclasses of Čech-analytic spaces, Comment. Math. Univ. Carolin., 25, 368–370.

    Google Scholar 

  11. Frolík, Z., (1985). On restrictions of projections along separable metric spaces, Report of the Department of Mathematics and Informatics 85-42, Delft University of Technology, Delft.

  12. Frolík, Z. and Holický, P., (1981). Decomposability of completely Suslin-additive families, Proc. Amer. Math. Soc., 82, 3, 359–365. DOI: 10.2307/2043940

    MATH  MathSciNet  Google Scholar 

  13. Frolík, Z. and Holický, P., (1983). Applications of Luzinian separation principles (non-separable case), Fund. Math., 117, 165–185.

    MATH  MathSciNet  Google Scholar 

  14. Frolík, Z. and Holický, P., (1985). Analytic and Luzin spaces (non-separable case), Topology Appl., 19, 129–156. DOI: 10.1016/0166-8641(85)90066-5

    Article  MATH  MathSciNet  Google Scholar 

  15. Hansell, R. W., (1971). Borel measurable mappings for nonseparable metric spaces, Trans. Amer. Math. Soc., 161, 145–169. DOI: 10.2307/1995934

    Article  MATH  MathSciNet  Google Scholar 

  16. Hansell, R. W., (1973). On the non-separable theory of k-Borel and k-Souslin sets, General Topology and Appl., 3, 161–195. DOI: 10.1016/0016-660X(73)90017-2

    Article  MathSciNet  Google Scholar 

  17. Hansell, R. W., (1974). On characterizing non-separable analytic and extended Borel sets as types of continuous images, Proc. Lond. Math. Soc., 28, 4, 683–699. DOI: 10.1112/plms/s3-28.4.683

    Article  MATH  MathSciNet  Google Scholar 

  18. Hansell, R. W., (1974). On Borel mappings and Baire functions, Trans. Amer. Math. Soc., 194, 195–211. DOI: 10.2307/1996801

    Article  MATH  MathSciNet  Google Scholar 

  19. Hansell, R. W., (1981). Point-finite Borel-additive families are of bounded class, Proc. Amer. Math. Soc., 83, 2, 375–378. DOI: 10.2307/2043532

    MATH  MathSciNet  Google Scholar 

  20. Hansell, R. W., (1983). Point-countable Souslin-additive families and σ-discrete reductions, in General Topology and its Relations to Modern Analysis and Algebra. V. (Proceedings of the Fifth Prague Topological Symposium, 1981), Sigma Series in Pure Mathematics, 3, Heldermann Verlag, Berlin, 254–260.

    Google Scholar 

  21. Hansell, R. W., (1986). F σ-set covers of analytic spaces and first class selectors, Proc. Amer. Math. Soc., 96, 2, 365–371. DOI: 10.2307/2046182

    MATH  MathSciNet  Google Scholar 

  22. Hansell, R. W., (1992). Descriptive topology, in Recent Progress in General Topology, M. Husek and J. van Mill eds., North-Holland, Amsterdam, 275–315.

    Google Scholar 

  23. Hansell, R. W., (2001). Descriptive sets and the topology of nonseparable Banach spaces, Serdica Math. J., 27, 1–66.

    MATH  MathSciNet  Google Scholar 

  24. Hansell, R. W.; Jayne, J. E. and Rogers, C. A., (1983). K-analytic sets, Mathematika, 30, 2, 189–221. DOI: 10.1112/S0025579300010524

    Article  MATH  MathSciNet  Google Scholar 

  25. Hansell, R. W.; Jayne, J. E. and Rogers, C. A., (1985). Separation of K-analytic sets, Mathematika, 32, 1, 147–190. DOI: 10.1112/S0025579300010962

    Article  MATH  MathSciNet  Google Scholar 

  26. Hansell, R. W.; Jayne, J. E. and Rogers, C. A., (1985). Piece-wise closed functions and almost discretely σ-decomposable families, Mathematika, 32, 2, 229–247. DOI: 10.1112/S0025579300011025

    Article  MATH  MathSciNet  Google Scholar 

  27. Hansell, R. W. and Pan, S., (1995). Perfect images of Čech-analytic spaces, Proc. Amer. Math. Soc., 123, 293–298. DOI: 10.2307/2160639

    MATH  MathSciNet  Google Scholar 

  28. Holický, P., (1991). Zdeněk Frolík and the descriptive theory of sets and spaces, Acta Univ. Carolin. Math. Phys., 32, 5–21.

    MATH  MathSciNet  Google Scholar 

  29. Holický, P., (1993). Čech analytic and almost K-descriptive spaces, Czechoslovak Math. J., 43, 451–466.

    MATH  MathSciNet  Google Scholar 

  30. Holický, P., (1994). Borel maps with the “point of continuity property” and completely Borel additive families in some nonmetrizable spaces, Proc. Amer. Math. Soc., 120, 3, 951–958. DOI: 10.2307/2160491

    Article  MATH  MathSciNet  Google Scholar 

  31. Holický, P., (1996). Luzin theorems for scattered-K-analytic spaces and Borel measures and Borel measures on them, Atti Semin. Mat. Fis. Univ. Modena Reggio Emilia, XLIV, 395–413.

    Google Scholar 

  32. Holický, P., (2001). Generalized analytic spaces, completeness and fragmentability, Czechoslovak Math. J., 51, 791–818.

    Article  MATH  MathSciNet  Google Scholar 

  33. Holický, P., (2004) Extensions of Borel measurable maps and ranges of Borel bimeasurable maps, Bull. Pol. Acad. Sci. Math., 52, 2, 151–167. DOI: 10.4064/ba52-2-6

    Article  MATH  MathSciNet  Google Scholar 

  34. Holický, P., (2006). Borel sets with countable sections for nonseparable spaces, Proc. Amer. Math. Soc., 134, 1519–1525. DOI: 10.1090/S0002-9939-05-08099-8

    Article  MATH  MathSciNet  Google Scholar 

  35. Holický, P., (2008). Borel sets with σ-compact sections for nonseparable spaces, Fund. Math., 199, 2, 139–154. DOI: 10.4064/fm199-2-4

    Article  MATH  MathSciNet  Google Scholar 

  36. Holický, P., (2008). Decompositions of Borel bimeasurable mappings between complete metric spaces, Topology Appl., 156, 2, 217–226. DOI: 10.1016/j.topol.2008.06.008

    Article  MATH  MathSciNet  Google Scholar 

  37. Holický, P. and Komínek, V., (2007). Descriptive properties of mappings of nonseparable Luzin spaces, Czechoslovak Math. J., 57, 1, 201–224. DOI: 10.1007/s10587-007-0056-6

    Article  MATH  MathSciNet  Google Scholar 

  38. Holický, P. and Pelant, J., (2004). Internal descriptions of absolute Borel classes, Topology Appl., 141, 87–104. DOI: 10.1016/j.topol.2003.10.004

    MATH  MathSciNet  Google Scholar 

  39. Holický, P. and Spurný, J., (2003). Perfect images of absolute Souslin and absolute Borel Tychonoff spaces, Topology Appl., 131, 281–294. DOI: 10.1016/S0166-8641(02)00356-5

    Article  MATH  MathSciNet  Google Scholar 

  40. Holický, P. and Spurný, J., (2004). F σ-mappings and the invariance of absolute Borel classes, Fund. Math., 182, 3, 193–204. DOI: 10.4064/fm182-3-1

    Article  MATH  MathSciNet  Google Scholar 

  41. Holický, P. and Zelený, M., (2000). A converse of the Arsenin-Kunugui theorem on Borel sets with σ-compact sections, Fund. Math., 165, 191–202.

    MATH  MathSciNet  Google Scholar 

  42. Jayne, J. E., (1976). Structure of analytic Hausdorff spaces, Mathematika, 23, 2, 208–211. DOI: 10.1112/S0025579300008809

    Article  MATH  MathSciNet  Google Scholar 

  43. Jayne, J. E.; Namioka, I. and Rogers, C. A., (1993). Topological properties of Banach spaces, Proc. Lond. Math. Soc, 66, 651–672. DOI: 10.1112/plms/s3-66.3.651

    Article  MATH  MathSciNet  Google Scholar 

  44. Jayne, J. E. and Rogers, C. A., (1980). K-analytic sets, Analytic Sets, Academic Press, London, 1–181.

    MATH  Google Scholar 

  45. Jayne, J. E. and Rogers, C. A., (1981). Piece-wise closed functions, Math. Ann., 255, 4, 499–518. DOI: 10.1007/BF01451930

    Article  MATH  MathSciNet  Google Scholar 

  46. Jayne, J. E. and Rogers, C. A., (1982). First level Borel functions and isomorphisms, J. Math. Pures Appl., 61, 177–205.

    MATH  MathSciNet  Google Scholar 

  47. Kaniewski, J. and Pol, R., (1975). Borel-measurable selectors for compact-valued mappings in the nonseparable case, Bull. Acad. Polon. Sci., 23, 1043–1050.

    MathSciNet  Google Scholar 

  48. Kechris, A. S., (1995). Classical Descriptive Set Theory, Springer-Verlag, New York.

    MATH  Google Scholar 

  49. Kuratowski, K., (1966). Topology, Vol. I, Academic Press, New York.

    Google Scholar 

  50. Louveau, A., (1980). A separation theorem for Σ 11 sets, Trans. Amer. Math. Soc., 260, 363–378. DOI: 10.2307/1998008

    MathSciNet  Google Scholar 

  51. Louveau, A., (1980). Ensembles analytiques et boréliens dans les espaces produits, Astérisque, 78.

  52. Marciszewski, W. and Pol, R., (2009). On Banach spaces whose norm-open sets are F σ-sets in the weak topology, J. Math. Anal. Appl., 350, 2, 708–722. DOI: 10.1016/j.jmaa.2008.06.007

    Article  MATH  MathSciNet  Google Scholar 

  53. Michael, E., (1982). On maps related to σ-locally finite and σ-discrete collections of sets, Pacific J. Math., 98, 139–152.

    MATH  MathSciNet  Google Scholar 

  54. Moschovakis, Y. N., (1980). Descriptive Set Theory, North-Holland, Amsterdam.

  55. Oncina, L., (2000). The JNR property and the Borel structure of a Banach space, Serdica Math. J., 26, 13–32.

    MATH  MathSciNet  Google Scholar 

  56. Oncina, L. and Raja, M., (2004). Descriptive compact spaces and renorming, Studia Math., 165, 39–52. DOI: 10.4064/sm165-1-3

    Article  MATH  MathSciNet  Google Scholar 

  57. Preiss, D., (1974). Completely additive disjoint system of Baire sets is of bounded class, Comment. Math. Univ. Carolin., 15, 341–344.

    MATH  MathSciNet  Google Scholar 

  58. Raja, M., (2002). On some class of Borel measurable maps and absolute Borel topological spaces, Topology Appl., 123, 2, 267–282. DOI: 10.1016/S0166-8641(01)00201-2

    Article  MATH  MathSciNet  Google Scholar 

  59. Saint-Raymond, J., (1976). Boréliens á coupes ϰσ, Bull. Soc. Math. France, 104, 389–406.

    MATH  MathSciNet  Google Scholar 

  60. Spurný, J., (2007). G δ-additive families in absolute Souslin spaces and Borel measurable selectors, Topology Appl., 154, 15, 2779–2785. DOI: 10.1016/j.topol.2007.05.012

    Article  MATH  MathSciNet  Google Scholar 

  61. Spurný, J., (2008). F σ-additive covers of Čech complete and scattered-K-analytic spaces, Fund. Math., 199, 131–138. DOI: 10.4064/fm199-2-3

    Article  MATH  MathSciNet  Google Scholar 

  62. Spurný, J. and Zelený , M., Additive families of low Borel classes and Borel measurable selectors, to appear in Canad. Math. Bull.

  63. Srivastava, S. M., (1998). A Course on Borel Sets, Springer-Verlag, New York.

    MATH  Google Scholar 

  64. Talagrand, M., (1975). Sur une conjecture de H. H. Corson, Bull. Sci. Math., 99, 211–212.

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Faculty of Mathematics and Physics, Department of Mathematical Analysis, Charles University, Sokolovská 83, 186 75, Prague 8, Czech Republic

    Petr Holický

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  1. Petr Holický
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Correspondence to Petr Holický.

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Dedicated to Professor Manuel Valdivia on the occasion of his 80th birthday

Submitted by Stanimir Troyanski

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Holický, P. Descriptive classes of sets in nonseparable spaces. RACSAM 104, 257–282 (2010). https://doi.org/10.5052/RACSAM.2010.17

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