# Lindelöf property and the iterated continuous function spaces

Fundamenta Mathematicae (1993)

- Volume: 143, Issue: 1, page 87-95
- ISSN: 0016-2736

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topSokolov, G.. "Lindelöf property and the iterated continuous function spaces." Fundamenta Mathematicae 143.1 (1993): 87-95. <http://eudml.org/doc/211994>.

@article{Sokolov1993,

abstract = {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.},

author = {Sokolov, G.},

journal = {Fundamenta Mathematicae},

keywords = {Lindelöf property},

language = {eng},

number = {1},

pages = {87-95},

title = {Lindelöf property and the iterated continuous function spaces},

url = {http://eudml.org/doc/211994},

volume = {143},

year = {1993},

}

TY - JOUR

AU - Sokolov, G.

TI - Lindelöf property and the iterated continuous function spaces

JO - Fundamenta Mathematicae

PY - 1993

VL - 143

IS - 1

SP - 87

EP - 95

AB - 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.

LA - eng

KW - Lindelöf property

UR - http://eudml.org/doc/211994

ER -

## References

top- [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 ${c}_{0}\left(\Gamma \right)$, Bull. Acad. Polon. Sci. 32 (1984), 681-688. Zbl0571.54014
- [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. Zbl0389.54009
- [13] R. Pol, A function space C(X) which is weakly Lindelöf but not weakly compactly generated, Studia Math. 64 (1979), 279-284. Zbl0424.46011
- [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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