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The Density Character of the Space \(C_p(X)\)

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Descriptive Topology and Functional Analysis

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 80))

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Abstract

The main purpose of this survey is to introduce to the reader the adequate framework and motivation for the recent results obtained relating the density character and the space of the continuous functions, [16]. The interest in this cardinal function has been continuous over the years. We will offer a vision of the process along the time and we will point out different general results. Specially, we are interested in those in which the space of the continuous functions appears as well as those in which duality plays an important role. Of course, precise classes of spaces are considered in each case to apply the results, which will take us forward to expose a parallel development and description of a specific class, in fact it will be the development of a different cardinal function, the number of Nagami, which measures the specific property of the space what makes things work well.

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References

  1. Arkhangel’ski\(\breve{1}\), A.V.: Topological Function Spaces, Mathematics and Its Applications (Soviet Series), vol. 78. Kluwer, Dordrecht (1992)

    Google Scholar 

  2. Amir, D., Lindenstrauss, J.: The structure of weakly compact sets in Banach spaces. Ann. Math. 88, 35–46 (1968)

    Article  MATH  MathSciNet  Google Scholar 

  3. Avilés, A.: The number of weakly compact sets which generate a Banach space. Israel J. Math. 159, 189–204 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  4. Canela, M.A.: \(K\)-analytic locally convex spaces. Port. Math. 41, 105–117 (1982)

    MATH  MathSciNet  Google Scholar 

  5. Cascales, B.: On \(K\)-analytic locally convex spaces. Arch. Math. 49, 232–244 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  6. Cascales, C., Muñoz, M., Orihuela, J.: The number of \(K\)-determination of topological spaces. RACSAM 106, 341–357 (2012)

    Article  MATH  Google Scholar 

  7. Cascales, C., Oncina, L.: Compactoid filters and USCO maps. J. Math. Anal. Appl. 282, 826–845 (2003)

    Google Scholar 

  8. Cascales, B., Orihuela, J.: On compactness in locally convex spaces. Math. Z. 195, 365–381 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  9. Cascales, B., Orihuela, J.: On pointwise and weak compactness in spaces of continuous functions. Bull. Soc. Math. Belg. Ser. B 40, 331–352 (1988)

    MATH  MathSciNet  Google Scholar 

  10. Choquet, G.: Theory of capacities. Ann. Inst. Fourier 5, 131–297 (1953)

    Article  MathSciNet  Google Scholar 

  11. Choquet, G.: Ensembles \(K\)-analytiques et \(K\)-sousliniens. Cas général et cas métrique. Ann. Inst. Fourier 9, 75–89 (1959)

    Article  MATH  MathSciNet  Google Scholar 

  12. Dunford, N., Schwartz, J.T.: Linear Operators. Part I. Interscience Pub, New York (1958)

    MATH  Google Scholar 

  13. Engelking, R.: General Topology. PWN-Polish Scientific Publishers, Warsaw (1977)

    MATH  Google Scholar 

  14. Fabian, M.: Gâteaux Differentiability of Convex Functions and Topology. Weak Asplund Spaces. Canadian Mathematical Society Series of Monographs and Advanced Texts. John Wiley and Sons, New York (1997)

    Google Scholar 

  15. Fabian, M., Habala, P., Hájek, P., Montesinos, V., Pelant, J., Zizler, V.: Functional Analysis and Infinite Dimensional Geometry. CMS Books in Mathematics, Springer, New York (2001)

    Book  MATH  Google Scholar 

  16. Ferrando, J.C., Ka̧kol, J., López-Pellicer, M., Muñoz Guillermo, M.: Some cardinal inequalities for spaces \(C_p(X)\). Topology Appl. 160, 1102–1107 (2013)

    Google Scholar 

  17. Fremlin, D.H.: \(K\)-analytic spaces with metrizable compacts. Mathematika 24, 257–261 (1977)

    Article  MathSciNet  Google Scholar 

  18. Hewitt, E.: A problem of set-theoretic topology. Duke Math. J. 10, 309–333 (1943)

    Article  MATH  MathSciNet  Google Scholar 

  19. Hewitt, E.: A remark on density characters. Bull. Amer. Math. Soc. 52, 641–643 (1946)

    Article  MATH  MathSciNet  Google Scholar 

  20. Hödel, R.: On a theorem of Arkhangel’skiĭ concerning Lindelöf \(p\)-spaces. Can. J. Math. 27, 459–468 (1975)

    Article  Google Scholar 

  21. Hödel, R.: Cardinal Functions I. In: Kunen, K., Vaughan, J.E. (eds.) Handbook of Set-Theoretic Topology, pp. 1–61. Elsevier, Amsterdam (1984)

    Google Scholar 

  22. Hunter, R.J., Lloyd, J.: Weakly compactly generated locally convex spaces. Proc. Math. Proc. Cambridge Philos. Soc. 82, 85–98 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  23. Jarchow, H.: Locally Convex Spaces. B.G. Teubner, Stuttgart (1981)

    Book  MATH  Google Scholar 

  24. Ka̧kol, J., Kubiś, W., López-Pellicer, M.: Descriptive Topology in Selected Topics of Functional Analysis. Springer, New York (2010)

    Google Scholar 

  25. Kelley, J.L.: General Topology. Springer, New York (1975)

    MATH  Google Scholar 

  26. Lindenstrauss, J.: Weakly compact sets: their topological properties and the Banach spaces they generate. Proc. Int. Symp. Topology Ann. Math. Stud. 69, 235–273 (1972)

    MathSciNet  Google Scholar 

  27. McCoy, R.A., Ntantu, I.: Topological Properties of Spaces of Continuous Functions. Springer, Berlin (1980)

    Google Scholar 

  28. Muñoz, M.: Bounds using the index of Nagami. Rocky Mountain J. Math. 41, 1977–1986 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  29. Nagami, K.: \(\Sigma \)-spaces. Fund. Math. 61, 169–192 (1969)

    MathSciNet  Google Scholar 

  30. Noble, N.: The density character of function spaces. Proc. Amer. Math. Soc. 42, 228–233 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  31. Orihuela, J.: Pointwise compactness in spaces of continuous functions. J. London Math. Soc. s2–36, 143–152 (1987)

    Google Scholar 

  32. Pondiczery, E.S.: Power problems in abstract spaces. Duke Math. J. 11, 835–837 (1944)

    Article  MATH  MathSciNet  Google Scholar 

  33. Pospíšil, B.: Sur la puissance d’un espace contenant une partie dense de puissance donnée. Časopis pro pěstování Matematiky a Fysiky 67, 89–96 (1937–1938)

    Google Scholar 

  34. Rosenthal, H.P.: The heredity problem for weakly compactly generated Banach spaces. Comp. Math. 124, 83–111 (1970)

    Google Scholar 

  35. Talagrand, M.: Espaces de Banach faiblement \(K\)-analytiques. Ann. Math. 110, 407–438 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  36. Talagrand, M.: Sur la \(K\)-analyticite de certains espaces d’operateurs. Israel J. Math. 32, 124–130 (1979)

    Google Scholar 

  37. Tkachuk, V.: Lindelöf \(\Sigma \)-spaces: an omnipresent class. RACSAM 104, 221–244 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  38. Uspenskiĭ, V.V.: A characterization of realcompactness in terms of the topology of pointwise convergence on the function space. Comm. Math. Univ. 24, 121–126 (1983)

    Google Scholar 

  39. Valdivia, M.: Topics in Locally Convex Spaces. Mathematics Studies, vol. 67. North-Holland Publishing, Amsterdam (1982)

    Google Scholar 

  40. Vašák, L.: On a generalization of weakly compactly generated Banach spaces. Studia Math. 70, 11–19 (1981)

    MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Departamento de Matemática Aplicada y Estadística, Universidad Politécnica de Cartagena, 30202, Cartagena, Murcia, Spain

    María Muñoz Guillermo

  2. Centro de Investigación Operativa, Universidad Miguel Hernández, 03202, Elche, Alicante, Spain

    J. C. Ferrando

  3. Departmento de Matemática Aplicada and IUMPA, Universitat Politècnica de València, 46002, Valencia, Spain

    M. López-Pellicer

Authors
  1. María Muñoz Guillermo
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  2. J. C. Ferrando
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  3. M. López-Pellicer
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Corresponding author

Correspondence to María Muñoz Guillermo .

Editor information

Editors and Affiliations

  1. Center Operations Research Univ. Inst., Universidad Miguel Hernandez, Elche, Alicante, Spain

    Juan Carlos Ferrando

  2. Matematica Aplicada, Universidad Politecnica de Valencia, Valencia, Valencia, Spain

    Manuel López-Pellicer

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Muñoz Guillermo, M., Ferrando, J.C., López-Pellicer, M. (2014). The Density Character of the Space \(C_p(X)\) . In: Ferrando, J., López-Pellicer, M. (eds) Descriptive Topology and Functional Analysis. Springer Proceedings in Mathematics & Statistics, vol 80. Springer, Cham. https://doi.org/10.1007/978-3-319-05224-3_4

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