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The strong Pytkeev property for topological groups and topological vector spaces

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Abstract

Generalizing the notion of metrisability, recently Tsaban and Zdomskyy (Houst J Math 35:563–571, 2009) introduced the strong Pytkeev property and proved a result stating that the space \(C_{c}(X)\) has this property for any Polish space \(X\). We show that the strong Pytkeev property for general topological groups is closely related to the notion of a \({\mathfrak {G}}\)-base, investigated in Gabriyelyan et al. (On topological groups with a small base and metrizability, preprint) and Ka̧kol et al. (Descriptive Topology in Selected Topics of Functional Analysis. Developments in Mathematics. Springer, Berlin, 2011). Our technique leads to an essential extension of Tsaban–Zdomskyy’s result. In particular, we prove that for a Čech-complete \(X\) the space \(C_{c}(X)\) has the strong Pytkeev property if and only if \(X\) is Lindelöf. We study the strong Pytkeev property for several well known classes of locally convex spaces including \((DF)\)-spaces and strict \((LM)\)-spaces. Strengthening results from Cascales et al. (Proc Am Math Soc 131:3623–3631, 2003) and Dudley (Proc Am Math Soc 27:531–534, 1971) we deduce that the space of distributions \({\mathfrak {D}}'(\varOmega )\) (which is not a \(k\)-space) has the strong Pytkeev property. We also show that any topological group with a \({\mathfrak {G}}\)-base which is a \(k\)-space has already the strong Pytkeev property. We prove that, if \(X\) is an \({\mathcal {MK}}_\omega \)-space, then the free abelian topological group \(A(X)\) and the free locally convex space \(L(X)\) have the strong Pytkeev property. We include various (counter) examples and pose a dozen open questions.

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References

  1. Arhangel’skii, A.V.: Topological Function Spaces (Translated from Russian). Kluwer Academic, Dordrecht (1992)

    Book  Google Scholar 

  2. Banakh, T.: Topological classification of strong duals to nuclear \((LF)\)-spaces. Studia Math. 138, 201–208 (2000)

    MathSciNet  MATH  Google Scholar 

  3. Banakh T.: Pytkeev \(\aleph _0\)-spaces. arXiv:1311.1468

  4. Banakh, T., Zdomskyy, L.: The topological structure of (homogeneous) spaces and groups with countable \(cs^\ast \)-character. Appl. Gen. Topol. 5, 25–48 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cascales, B.:, Ka̧kol, J., Saxon, S.A.: Metrizability vs. Fréchet-Urysohn property. Proc. Am. Math. Soc. 131, 3623–3631 (2003)

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

    Article  MathSciNet  MATH  Google Scholar 

  7. Christensen, J.P.R.: Topology and Borel Structure. In: Descriptive topology and set theory with applications to functional analysis and measure theory. North-Holland Mathematical Studies, vol. 10, pp iii+133, North-Holland Publishing Co., Amsterdam, London (1974)

  8. Day, M.M.: Normed Linear Spaces. Springer, Berlin (1958)

    Book  MATH  Google Scholar 

  9. Dudley, R.M.: Convergence of sequences of distributions. Proc. Am. Math. Soc. 27, 531–534 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  10. Engelking, R.: General Topology. Panstwowe Wydawnictwo Naukowe, Waszawa (1977)

    MATH  Google Scholar 

  11. Ferrando, J.C., Ka̧kol, J.: On precompact sets in spaces \(C_{c}(X)\). Georg. Math. J. (preprint)

  12. Ferrando, J.C., Ka̧kol, J., López Pellicer, M., Saxon, S.A.: Tightness and distinguished Fréchet spaces. J. Math. Anal. Appl. 324, 862–881 (2006)

  13. Flood, J.: Free Topological vector spaces. Ph.D. thesis. Australian National University, Canberra (1975)

  14. Flood, J.: Free locally convex spaces. Dissertationes Math CCXXI, PWN, Warczawa (1984)

  15. Gabriyelyan, S.S.: Groups of quasi-invariance and the Pontryagin duality. Topol. Appl. 157, 2786–2802 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. Gabriyelyan, S.S.: Topologies on groups determined by sets of convergent sequences. J. Pure Appl. Algebra 217, 786–802 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  17. Gabriyelyan, S.S., Ka̧kol, J., Leiderman, A.: On topological groups with a small base and metrizability (preprint)

  18. Graev, M.: Free topological groups. Izv. Akad. Nauk SSSR Ser. Mat. 12, 278–324 (1948) (In Russian). Topology and Topological Algebra. Translation Series 1, 8, 305–364 (1962)

  19. Gruenhage, G.: Generalized Metric Spaces. Handbook of Set-Theoretic Topology, pp. 423–501. North-Holland, Amsterdam (1984)

  20. Gruenhage, G., Tsaban, B., Zdomskyy, L.: Sequential properties of function spaces with the compact-open topology. Topol. Appl. 158, 387–391 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  21. Ka̧kol, J., Kubiś, W., Lopez-Pellicer, M.: Descriptive Topology in Selected Topics of Functional Analysis. Developments in Mathematics. Springer, Berlin (2011)

  22. Ka̧kol, J., Saxon, S.A.: Montel (DF)-spaces, sequential (LM)-spaces and the strongest locally convex topology. J. Lond. Math. Soc. 66, 388–406 (2002)

  23. Malykhin, V.I., Tironi, G.: Weakly Fréchet–Urysohn and Pytkeev spaces. Topol. Appl. 104, 181–190 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  24. Markov, A.A.: On free topological groups. Dokl. Akad. Nauk SSSR 31, 299–301 (1941)

    MATH  Google Scholar 

  25. McCoy, R.A.: Function spaces which are \(k\)-spaces. Topol. Proc. 5, 139–146 (1980)

    MathSciNet  Google Scholar 

  26. McCoy, R.A., Ntantu, I.: Topological Properties of Spaces of Continuous Functions. Lecture Notes in Mathematics, vol. 1315, pp iv+124, Springer, Berlin (1988)

  27. Meise, R., Vogt, D.: Introduction to Functional Analysis. Oxford Graduate Text in Mathematics, vol 2, pp x+437, The Clarendon Press, Oxford University Press, New York (1997)

  28. Nyikos, P.J.: Metrizability and Fréchet–Urysohn property in topological groups. Proc. Am. Math. Soc. 83, 793–801 (1981)

    MathSciNet  MATH  Google Scholar 

  29. Pérez Carreras, P., Bonet, J.: Barrelled Locally Convex Spaces. North-Holland Mathematics Studies, vol. 131. North-Holland, Amsterdam (1987)

  30. Pol, R.: Normality in function spaces. Fund. Math. 84, 145–155 (1974)

    MathSciNet  MATH  Google Scholar 

  31. Protasov, I.V., Zelenyuk, E.G.: Topologies on Groups Determined by Sequences. Monograph Series, Mathematical Studies. VNTL, L’viv (1999)

  32. Pytkeev, E.G.: Sequentiality of spaces of continuous functions. (Russian) Uspekhi Mat. Nauk 37, 197–198 (1982)

    MathSciNet  Google Scholar 

  33. Pytkeev, E.G.: On maximally resolvable spaces. Proc. Steklov Inst. Math. 154, 225–230 (1984)

    MATH  Google Scholar 

  34. Pytkeev, E.G.: Spaces of continuous and Baire functions in weak topologies. Doktor Sci. Dissertation, Ekaterinburg (in Russian) (1993)

  35. Raĭkov, D.A.: Free locally convex spaces for uniform spaces. Math. Sb. 63, 582–590 (1964)

    Google Scholar 

  36. Sakai, M.: Function spaces with a countable \(cs^\ast \)-network at a point. Topol. Appl. 156, 117–123 (2008)

    Article  MATH  Google Scholar 

  37. Schaefer, H.H.: Topological Vector Spaces. Macmillan, New York (1966)

    MATH  Google Scholar 

  38. Tkachenko, M.G.: On completeness of free abelian topological groups. Sov. Math. Dokl. 27, 341–345 (1983)

    MATH  Google Scholar 

  39. Tkachuk, V.V.: A space \(C_{p}(X)\) is dominated by irrationals if and only if it is \(K\)-analytic. Acta Math. Hung. 107, 253–265 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  40. Tsaban, B., Zdomskyy, L.: On the Pytkeev property in spaces of continuous functions (II). Houst. J. Math. 35, 563–571 (2009)

    MathSciNet  MATH  Google Scholar 

  41. Uspenskiĭ, V.V.: On the topology of free locally convex spaces. Sov. Math. Dokl. 27, 781–785 (1983)

    Google Scholar 

  42. Uspenskiĭ, V.V.: Free topological groups of metrizable spaces. Math. USSR-Izv. 37, 657–680 (1991)

    Article  MathSciNet  Google Scholar 

  43. Vogt, D.: Distinguished Köthe spaces. Math. Z. 202, 143–146 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  44. Valdivia, M.: Quasi-LB-spaces. J. Lond. Math. Soc. 35, 149–168 (1987)

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

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Acknowledgments

The authors would like to thank the referees for valuable remarks and suggestions.

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Authors and Affiliations

  1. Department of Mathematics, Ben-Gurion University of the Negev, Beer Sheva, Israel

    S. S. Gabriyelyan & A. Leiderman

  2. Faculty of Mathematics and Informatics, A. Mickiewicz University, 61-614 , Poznan, Poland

    J. Ka̧kol

Authors
  1. S. S. Gabriyelyan
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  2. J. Ka̧kol
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  3. A. Leiderman
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Corresponding author

Correspondence to A. Leiderman.

Additional information

Communicated by S.-D. Friedman.

Research of the J. Ka̧kol is supported by National Center of Science, Poland, Grant No. N N201 605340 and also by Generalitat Valenciana, Conselleria d’Educació, Cultura i Esport, Spain, Grant PROMETEO/2013/058.

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Gabriyelyan, S.S., Ka̧kol, J. & Leiderman, A. The strong Pytkeev property for topological groups and topological vector spaces. Monatsh Math 175, 519–542 (2014). https://doi.org/10.1007/s00605-014-0639-x

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  • DOI: https://doi.org/10.1007/s00605-014-0639-x

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