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1993 | 143 | 1 | 87-95
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Lindelöf property and the iterated continuous function spaces

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We give an example of a compact space X whose iterated continuous function spaces $C_{p}(X)$, $C_pC_p(X), ...$ are Lindelöf, but X is not a Corson compactum. This solves a problem of Gul'ko (Problem 1052 in [11]). We also provide a theorem concerning the Lindelöf property in the function spaces $C_{p}(X)$ on compact scattered spaces with the $ω_1$th derived set empty, improving some earlier results of Pol [12] in this direction.
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  • Faculty of Mechanics and Mathematics, Tomsk State University, Pr. Lenina 36, 634010 Tomsk, Russia
  • [1] A. V. Arkhangel'skiĭ, Topological Function Spaces, Moscow Univ. Press, 1989 (in Russian); English transl.: Kluwer Acad. Publ., Dordrecht 1992.
  • [2] K. Ciesielski and R. Pol, A weakly Lindelöf function space C(K) without any continuous injection into $c_0(Γ)$, Bull. Acad. Polon. Sci. 32 (1984), 681-688.
  • [3] W. G. Fleissner, Applications of stationary sets in topology, in: Surveys in General Topology, Academic Press, 1980, 163-193.
  • [4] S. P. Gul'ko, On properties of subsets of Σ-products, Dokl. Akad. Nauk SSSR 237 (1977), 505-508 (in Russian); English transl.: Soviet Math. Dokl. 18 (1977), 1438-1442.
  • [5] S. P. Gul'ko, On properties of some function spaces, Dokl. Akad. Nauk SSSR 243 (1978), 839-842 (in Russian); English transl.: Soviet Math. Dokl. 19 (1978), 1420-1424.
  • [6] S. P. Gul'ko, On properties of function spaces, in: Seminar on General Topology, Moscow Univ. Press, 1981, 8-41 (in Russian).
  • [7] T. Jech, Set Theory, Academic Press, New York 1978.
  • [8] V. I. Malykhin, On the tightness and the Suslin number of exp X and of a product of spaces, Dokl. Akad. Nauk SSSR 203 (1972), 1001-1003 (in Russian); English transl.: Soviet Math. Dokl. 13 (1972), 496-499.
  • [9] S. Negrepontis, Banach spaces and topology, in: Handbook of Set-Theoretic Topology, North-Holland, Amsterdam 1984, 1045-1142.
  • [10] O. G. Okunev, On the weak topology of conjugate spaces and the t-equivalence relation, Mat. Zametki 46 (1989), 53-59 (in Russian).
  • [11] Open Problems in Topology, J. van Mill and G. M. Reed (eds.), North-Holland, Amsterdam 1990.
  • [12] R. Pol, Concerning function spaces on separable compact spaces, Bull. Acad. Polon. Sci. 25 (1977), 993-997.
  • [13] R. Pol, A function space C(X) which is weakly Lindelöf but not weakly compactly generated, Studia Math. 64 (1979), 279-284.
  • [14] Z. Semadeni, Banach Spaces of Continuous Functions, PWN, Warszawa 1971.
  • [15] O. V. Sipachova, The structure of iterated function spaces in the topology of pointwise convergence for Eberlein compacta, Mat. Zametki 47 (1990), 91-99 (in Russian).
  • [16] G. A. Sokolov, On Lindelöf spaces of continuous functions, ibid. 36 (1986), 887-894 (in Russian).
  • [17] E. A. Reznichenko, Convex and compact subsets of function spaces and locally convex spaces, Ph.D. thesis, Moscow Univ., 1992 (in Russian).
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