A characterization of 𝑋 for which spaces 𝐶_{𝑝}(𝑋) are distinguished and its applications

Ka̧kol, Jerzy; Leiderman, Arkady

doi:10.1090/bproc/76

A characterization of $X$ for which spaces $C_p(X)$ are distinguished and its applications
HTML articles powered by AMS MathViewer

by Jerzy Ka̧kol and Arkady Leiderman HTML | PDF

Proc. Amer. Math. Soc. Ser. B 8 (2021), 86-99

Abstract:

We prove that the locally convex space $C_{p}(X)$ of continuous real-valued functions on a Tychonoff space $X$ equipped with the topology of pointwise convergence is distinguished if and only if $X$ is a $\Delta$-space in the sense of Knight in [Trans. Amer. Math. Soc. 339 (1993), pp. 45–60]. As an application of this characterization theorem we obtain the following results:

[1)] If $X$ is a Čech-complete (in particular, compact) space such that $C_p(X)$ is distinguished, then $X$ is scattered.
[2)] For every separable compact space of the Isbell–Mrówka type $X$, the space $C_p(X)$ is distinguished.
[3)] If $X$ is the compact space of ordinals $[0,\omega _1]$, then $C_p(X)$ is not distinguished.

We observe that the existence of an uncountable separable metrizable space $X$ such that $C_p(X)$ is distinguished, is independent of ZFC. We also explore the question to which extent the class of $\Delta$-spaces is invariant under basic topological operations.

References

K. Alster, Some remarks on Eberlein compacts, Fund. Math. 104 (1979), no. 1, 43–46. MR 549380, DOI 10.4064/fm-104-1-43-46
A. V. Arkhangel’skiĭ, Topological Function Spaces, Kluwer, Dordrecht, 1992.
Taras Banakh and Arkady Leiderman, Uniform Eberlein compactifications of metrizable spaces, Topology Appl. 159 (2012), no. 7, 1691–1694. MR 2904055, DOI 10.1016/j.topol.2011.06.060
A. I. Bashkirov, On continuous maps of Isbell spaces and strong $0$-dimensionality, Bull. Acad. Polon. Sci. Sér. Sci. Math. 27 (1979), no. 7-8, 605–611 (1980) (English, with Russian summary). MR 581560
Murray Bell and Witold Marciszewski, On scattered Eberlein compact spaces, Israel J. Math. 158 (2007), 217–224. MR 2342465, DOI 10.1007/s11856-007-0011-0
Klaus D. Bierstedt and José Bonet, Density conditions in Fréchet and (DF)-spaces, Rev. Mat. Univ. Complut. Madrid 2 (1989), no. suppl., 59–75. Congress on Functional Analysis (Madrid, 1988). MR 1057209
Klaus D. Bierstedt and José Bonet, Some aspects of the modern theory of Fréchet spaces, RACSAM. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 97 (2003), no. 2, 159–188 (English, with English and Spanish summaries). MR 2068172
H. G. Dales, F. K. Dashiell Jr., A. T.-M. Lau, and D. Strauss, Banach spaces of continuous functions as dual spaces, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, Springer, Cham, 2016. MR 3588279, DOI 10.1007/978-3-319-32349-7
Jean Dieudonné and Laurent Schwartz, La dualité dans les espaces $\scr F$ et $(\scr L\scr F)$, Ann. Inst. Fourier (Grenoble) 1 (1949), 61–101 (1950) (French). MR 38553
Ryszard Engelking, General topology, 2nd ed., Sigma Series in Pure Mathematics, vol. 6, Heldermann Verlag, Berlin, 1989. Translated from the Polish by the author. MR 1039321
Marián Fabian, Petr Habala, Petr Hájek, Vicente Montesinos Santalucía, Jan Pelant, and Václav Zizler, Functional analysis and infinite-dimensional geometry, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, vol. 8, Springer-Verlag, New York, 2001. MR 1831176, DOI 10.1007/978-1-4757-3480-5
Juan Carlos Ferrando and Jerzy Kąkol, Metrizable bounded sets in $C(X)$ spaces and distinguished $C_p(X)$ spaces, J. Convex Anal. 26 (2019), no. 4, 1337–1346. MR 4028412
Juan Carlos Ferrando, Jerzy Kąkol, and Stephen A. Saxon, Examples of nondistinguished function spaces $C_p(X)$, J. Convex Anal. 26 (2019), no. 4, 1347–1348. MR 4028413
J. C. Ferrando, J. Ka̧kol, A. Leiderman, and S. A. Saxon, Distinguished $C_p(X)$ spaces, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 115 (2021), no. 1, Paper No. 27, 18. MR 4182104, DOI 10.1007/s13398-020-00967-4
William G. Fleissner, Squares of $Q$ sets, Fund. Math. 118 (1983), no. 3, 223–231. MR 736282, DOI 10.4064/fm-118-3-223-231
William G. Fleissner and Arnold W. Miller, On $Q$ sets, Proc. Amer. Math. Soc. 78 (1980), no. 2, 280–284. MR 550513, DOI 10.1090/S0002-9939-1980-0550513-4
Alexandre Grothendieck, Sur les espaces ($F$) et ($DF$), Summa Brasil. Math. 3 (1954), 57–123 (French). MR 75542
F. Hernández-Hernández and M. Hrušák, Topology of Mrówka-Isbell spaces, Pseudocompact topological spaces, Dev. Math., vol. 55, Springer, Cham, 2018, pp. 253–289. MR 3822423
Gottfried Köthe, Topological vector spaces. II, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 237, Springer-Verlag, New York-Berlin, 1979. MR 551623
B. Knaster and K. Urbanik, Sur les espaces complets séparables de dimension $0$, Fund. Math. 40 (1953), 194–202 (French). MR 60221, DOI 10.4064/fm-40-1-194-202
R. W. Knight, $\Delta$-sets, Trans. Amer. Math. Soc. 339 (1993), no. 1, 45–60. MR 1196219, DOI 10.1090/S0002-9947-1993-1196219-6
Kenneth Kunen, Set theory, Studies in Logic and the Foundations of Mathematics, vol. 102, North-Holland Publishing Co., Amsterdam-New York, 1980. An introduction to independence proofs. MR 597342
Arnold W. Miller, Special subsets of the real line, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 201–233. MR 776624
Peter J. Nyikos, A history of the normal Moore space problem, Handbook of the history of general topology, Vol. 3, Hist. Topol., vol. 3, Kluwer Acad. Publ., Dordrecht, 2001, pp. 1179–1212. MR 1900271
Peter Nyikos and Juan J. Schäffer, Flat spaces of continuous functions, Studia Math. 42 (1972), 221–229. MR 308761, DOI 10.4064/sm-42-3-221-229
Teodor C. Przymusiński, The existence of $Q$-sets is equivalent to the existence of strong $Q$-sets, Proc. Amer. Math. Soc. 79 (1980), no. 4, 626–628. MR 572316, DOI 10.1090/S0002-9939-1980-0572316-7
T. C. Przymusiński, Normality and separability of Moore spaces, in: Set-Theoretic Topology, Academic Press, New York, 1977, 325–337.
G. M. Reed, On normality and countable paracompactness, Fund. Math. 110 (1980), no. 2, 145–152. MR 600588, DOI 10.4064/fm-110-2-145-152
Zbigniew Semadeni, Banach spaces of continuous functions. Vol. I, Monografie Matematyczne, Tom 55, PWN—Polish Scientific Publishers, Warsaw, 1971. MR 0296671
D. B. Shakhmatov, M. G. Tkačenko, V. V. Tkachuk, S. Watson, and R. G. Wilson, Neither first countable nor Čech-complete spaces are maximal Tychonoff connected, Proc. Amer. Math. Soc. 126 (1998), no. 1, 279–287. MR 1443164, DOI 10.1090/S0002-9939-98-04203-8
A. H. Stone, Kernel constructions and Borel sets, Trans. Amer. Math. Soc. 107 (1963), 58-70; errata, ibid. 107 (1963), 558. MR 0151935, DOI 10.1090/S0002-9947-1963-0151935-0
Jun Terasawa, Spaces $N\cup R$ need not be strongly $0$-dimensional, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astonom. Phys. 25 (1977), no. 3, 279–281 (English, with Russian summary). MR 0451214
M. L. Wage, W. G. Fleissner, and G. M. Reed, Normality versus countable paracompactness in perfect spaces, Bull. Amer. Math. Soc. 82 (1976), no. 4, 635–639. MR 410665, DOI 10.1090/S0002-9904-1976-14150-X