Abstract
The following results are obtained: (1) a criterion for the separability of the space of continuous functions C(X) with the set-open topology; (2) a criterion for the sequential separability of the space Cp(A|X), where A ⊆ X; (3) an answer to Velichko’s question of whether a set-theoretic condition on a metric space X in a criterion for the sequential separability of Cp(X) is necessary.
Article PDF
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
A. V. Osipov, “Topological-algebraic properties of function spaces with set-open topologies,” Topology Appl. 159 (3), 800–805 (2012).
A. V. Osipov, “On the completeness properties of the C-compact-open topology on C(X),” in Trudy Inst. Mat. iMekh. UrO RAN (2012), Vol. 18 (2), pp. 191–198 [in Russian].
A. V. Osipov, “Algebraic structures on the space of continuous maps,” Vestn. Tomsk. Gos. Univ.Mat.Mekh., No. 1, 47–53 (2012) [in Russian].
A. V. Osipov, “The dual of Crc(X),” The dual of, No. 1, 41–49 (2012) [in Russian].
A. V. Osipov, “The C-compact-open topology on function spaces,” Topology Appl. 159 (13), 3059–3066 (2012).
A. V. Osipov, “The set-open topology,” Topology Proc. 37, 205–217 (2011).
M. O. Asanov, Spaces of Continuous Mappings, Cand. Sci. (Phys.–Math.) Dissertation (Sverdlovsk, 1980) [in Russian].
A. V. Osipov and A. V. Kosolobov, “On sequentially compact-open topology,” Vestn. Udmurt. Univ., Mat. Mekh. Komp’yut. Nauki, No. 3, 75–84 (2011).
R. Engelking, General Topology (PWN—Polish Scientific Publishers,Warsaw, 1977;Mir, Moscow, 1986).
N. Noble, “The density character of function spaces,” Proc. Amer. Math. Soc. 42 (1), 228–233 (1974).
P. Gartside, J. T. H. Lo, and A. Marsh, “Sequential density,” Topology Appl. 130 (1), 75–86 (2003).
N. V. Velichko, “On sequential separability,” Mat. Zametki 78 (5), 652–657 (2005) [Math. Notes 78 (5), 610–614 (2005)].
K. Kuratowskii, Topology (Academic Press, New York–London, Panstwowe Wydawnictwo Naukowe, Warsaw, 1966; Mir, Moscow, 1966), Vol.1.
A. W. Miller, “On the length of Borel hierarchies,” Ann.Math. Logic 16 (3), 233–267 (1979).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © A.V. Osipov, 2018, published in Matematicheskie Zametki, 2018, Vol. 104, No. 1, pp. 87–98.
About this article
Cite this article
Osipov, A.V. Separability and Sequential Separability of the Space C(X). Math Notes 104, 86–95 (2018). https://doi.org/10.1134/S0001434618070106
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0001434618070106