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On Distinguished Spaces \(C_p(X)\) of Continuous Functions

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Functional Analysis and Continuous Optimization (IMFACO 2022)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 424))

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Abstract

This survey paper presents a number of recent results related to the study of spaces \(C_{p}\left( X\right) \) of continuous real-valued functions on a Tychonoff spaces X with the pointwise topology that are distinguished, i.e. spaces \(C_{p}\left( X\right) \) which are large subspaces of \(\mathbb {R}^{X}\), equivalently, spaces \(C_p(X)\) whose strong dual \(L_{\beta }\left( X\right) \) of \(C_p(X)\) carries the finest locally convex topology. Some open questions are included.

The research of the first named author is supported by the GAČR Project 20-22230L and RVO: 67985840. He thanks also the Center For Advanced Studies in Mathematics of Ben-Gurion University of the Negev for financial support during his visit in 2022.

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References

  1. Alster, K.: Some remarks on Eberlein compacts. Fund. Math. 104, 43–46 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  2. Arkhangel’skii, A.V.: \({C}_{p}\)-theory. In: Hušek, M., van Mill, J. (eds.) Recent progress in general topology, pp. 1–56. Elsevier, Amsterdam (1992)

    Google Scholar 

  3. Aviles, A., Guerrero Sanchez, D.: Are Eberlein-Grothendieck scattered spaces \(\sigma \)-discrete? Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 108, 849–859 (2014). https://doi.org/10.1007/s13398-013-0146-2

  4. Balogh, Z.: On compact Hausdorff spaces of countable tightness. Proc. Am. Math. Soc. 105, 755–764 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  5. Balogh, Z., Dow, A., Fremlin, D.H., Nyikos, P.J.: Countable tightness and proper forcing. Bull. Am. Math. Soc. (New Series) 19, 295–298 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bell, M., Marciszewski, W.: On scattered Eberlein compact spaces. Israel J. Math. 158, 217–224 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bierstedt, D., Bonet, J.: Dual density conditions in\((DF)\)-spaces I. Results Math. 14, 242–274 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bierstedt, D., Bonet, J.: Density conditions in Fréchet and \((DF)\)-spaces. Rev. Mat. Complut. 2, 59–75 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bierstedt, D., Bonet, J.: Some aspects of the modern theory of Fréchet spaces. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 97, 159–188 (2003)

    MathSciNet  MATH  Google Scholar 

  10. Engelking, R.: General Topology. Heldermann, Berlin (1989)

    MATH  Google Scholar 

  11. Fabian, M., Hájek, P., Montesinos, V., Zizler, V.: A quantitative version of Krein’s theorem. Rev. Mat. Iberoam. 21, 237–248 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ferrando, J.C., Ka̧kol J.: Metrizable bounded sets in \(C(X)\) spaces and distinguished \(C_{p}(X)\) spaces. J. Convex. Anal. 26, 1337–1346 (2019)

    Google Scholar 

  13. Ferrando, J.C., Ka̧kol, J., López-Pellicer, M.: A characterization of distinguished Fréchet spaces. Math. Nachr. 279, 1783–1786 (2006)

    Google Scholar 

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

    Google Scholar 

  15. Ferrando, J.C., Ka̧kol, J., Leiderman, A., Saxon, S.A.: Distinguished \(C_p(X)\) spaces. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 115(1), 27 (2021)

    Google Scholar 

  16. Ferrando, J.C., Saxon, S.A.: If not distinguished, is \(C_{p}(X)\) even close? Proc. Am. Math. Soc. 149, 2583–2596 (2021)

    Article  MATH  Google Scholar 

  17. Ferrando, J.C.: Approximation by pointwise bounded sets of continuous functions. Topol. Appl. 317, 108191 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  18. Filatova, M.A.: Resolvability of Lindelöf spaces. J. Math. Sci. (N.Y.) 146(1), 5603–5607 (2007)

    Google Scholar 

  19. Fleissner, W.M., Miller, A.W.: On \(Q\) sets. Proc. Am. Math. Soc. 78, 280–284 (1980)

    MATH  Google Scholar 

  20. Gruenhage, G.: Some results on spaces having an ortho-base or a base of subinfinite rank. Topol. Proc. 2, 151–159 (1977)

    MATH  Google Scholar 

  21. Heinrich, S.: Ultrapowers of locally convex spaces and applications I. Math. Nachr. 118, 285–315 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  22. Hodel, R.: Cardinal functions I. In: Kunen, K., Vaughan, J. (eds.) Handbook of Set-Theoretic Topology, pp. 1–61. North Holland, Amsterdam (1984)

    Google Scholar 

  23. Hrušák, M., Tamariz-Mascarúa, Á., Tkachenko, M. (eds.): Pseudocompact Topological Spaces, A Survey of Classic and New Results with Open Problems. Springer (2018)

    Google Scholar 

  24. Horváth, J.: Topological Vector Spaces and Distributions. Addison Wesley, Reading (1966)

    Google Scholar 

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

    Google Scholar 

  26. Juhasz, I., van Mill, J.: Countably compact spaces all countable subsets of which are scattered. Commentationes Mathematicae Universitatis Carolinae 22, 851–855 (1981)

    MathSciNet  MATH  Google Scholar 

  27. Juhász, I., Soukup, L., Szentmiklóssy, Z.: Regular spaces of small extent are \(\omega \) -resolvable. Fund. Math. 228, 27–46 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  28. Ka̧kol, J., Leiderman, A.: A characterization of \(X\) for which spaces \(C_p(X)\) are distinguished and its applications. Proc. Am. Math. Soc. Ser. B 8, 86–99 (2021)

    Google Scholar 

  29. Ka̧kol, J., Leiderman, A.: Basic properties of \(X\) for which the space \(C_p(X)\) is distinguished. Proc. Am. Math. Soc. Ser. B 8, 267–280 (2021)

    Google Scholar 

  30. Köthe, G.: Topological Vector Spaces I. Springer, Berlin (1983)

    Book  MATH  Google Scholar 

  31. Krawczyk, A., Marciszewski, W., Michalewski, H.: Remarks on the set of \(G_{\delta }\) -points in Eberlein and Corson compact spaces. Topol. Appl. 156, 1746–1748 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  32. Leiderman, A., Tkachenko, M.: Metrizable quotients of free topological groups. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 114, 124 (2020). https://doi.org/10.1007/s13398-020-00855-x

  33. Leiderman, A., Szeptycki, P.: On \(\Delta \) -spaces. Israel J. Math. (to appear)

    Google Scholar 

  34. Leiderman, A., Tkachuk, V.: Pseudocompact \(\Delta \)-spaces are often scattered. Monatshefte fur Mathematik 197, 493–503 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  35. Namioka, J., Phelps, R.R.: Banach spaces which are Asplund spaces I. Duke Math. J. 42, 735–750 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  36. Narichi, L., Beckenstein, E.: Topological Vector Spaces, 2nd edn. Chapman and Hall/CRC (2010)

    Google Scholar 

  37. Peris, A.: Quasinormable spaces and the problem of topologies of Grothendieck. Ann. Acad. Sci. Fenn. Math. 19, 167–203 (1994)

    MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  39. Przymusiński, T.C.: Normality and separability of Moore spaces. In: Set-Theoretic Topology, pp. 325–337. Academic Press, New York (1977)

    Google Scholar 

  40. Reed, G.M.: On normality and countable paracompactness. Fund. Math. 110, 145–152 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  41. Robertson, W.J., Saxon, S.A., Robertson, A.P.: Barrelled spaces and dense vector subspaces. Bull. Austral. Math. Soc. 37, 383–388 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  42. Saxon, S.A.: The density character of topological vector spaces. In: Lau, A., Tweddle, I. (eds.) Topological Vector Spaces, Algebras and Related Areas, vol. 316, pp. 24–36, Longman PRNM (1994)

    Google Scholar 

  43. Saxon, S.A., Sánchez Ruiz, L.M.: Optimal cardinals for metrizable barrelled spaces. J. Lond. Math. Soc. 51, 137–147 (1995)

    Google Scholar 

  44. Saxon, S.A., Tweddle, I.: The fit and flat components of barrelled spaces. Bull. Austral. Math. Soc. 51, 521–528 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  45. Semadeni, Z.: Banach Spaces of Continuous Functions, vol. I. PWN - Polish Scientific Publishers, Warszawa (1971)

    MATH  Google Scholar 

  46. Stone, A.H.: Kernel constructions and Borel sets. Trans. Am. Math. Soc. 107, 58–70 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  47. Taskinen, J.: Examples of non-distinguished Fréchet spaces. Ann. Acad. Sci. Fenn. Math. 14, 75–88 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  48. Tkachuk, V.V.: A \(C_{p}\)-theorem Problem Book. Compactness in Function Spaces. Springer (2015)

    Google Scholar 

  49. Tkachuk, V.V.: Lindelöf \(P\) -spaces need not be Sokolov. Math. Slovaca 67(1), 227–234 (2017)

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Faculty of Mathematics and Informatics, A. Mickiewicz University, 61-614, Poznań, Poland

    J. Ka̧kol

  2. Institute of Mathematics, Czech Academy of Sciences, Prague, Czechia

    J. Ka̧kol

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

    A. Leiderman

Authors
  1. J. Ka̧kol
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  2. A. Leiderman
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Corresponding author

Correspondence to J. Ka̧kol .

Editor information

Editors and Affiliations

  1. Center of Operations Research, Miguel Hernández University of Elche, Elche, Alicante, Spain

    José M. Amigó

  2. Center of Operations Research, Miguel Hernández University of Elche, Elche, Alicante, Spain

    María J. Cánovas

  3. Department of Mathematics, Alicante University, San Vicente del Raspeig, Alicante, Spain

    Marco A. López-Cerdá

  4. Institute for Pure and Applied Mathematics (IUMPA), Polytechnic University of Valencia, Valencia, Spain

    Manuel López-Pellicer

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Ka̧kol, J., Leiderman, A. (2023). On Distinguished Spaces \(C_p(X)\) of Continuous Functions. In: Amigó, J.M., Cánovas, M.J., López-Cerdá, M.A., López-Pellicer, M. (eds) Functional Analysis and Continuous Optimization. IMFACO 2022. Springer Proceedings in Mathematics & Statistics, vol 424. Springer, Cham. https://doi.org/10.1007/978-3-031-30014-1_10

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