Some properties of the quasicompact-open topology on C(X)

Tokat, Deniz¹ ; Osmanoğlu, İsmail¹

¹ Department of Mathematics, Faculty of Arts and Sciences, Nevsehir Hacı Bektas Veli University, 50300 Nevsehir, Turkey

Journal of nonlinear sciences and its applications, Tome 9 (2016) no. 6, p. 3511-3518.

Résumé

This paper introduces quasicompact-open topology on C(X) and compares this topology with the compact-open topology and the topology of uniform convergence. Then it examines submetrizability, metrizability, separability, and second countability of the quasicompact-open topology on C(X).

Détail

DOI : 10.22436/jnsa.009.06.06

Classification : 54C35, 54D65, 54E35
Keywords: Function space, set-open topology, compact-open topology, quasicompactness, separability, submetrizability, second countability.

Bibliographie

[1] Al-Ani, A. T. Countably z-compact spaces, Arch. Math. (Brno), Volume 50 (2014), pp. 97-100

[2] Arens, R. F. A topology for spaces of transformations, Ann. of Math., Volume 47 (1946), pp. 480-495

[3] Arens, R.; Dugundji, J. Topologies for function spaces, Pacific J. Math., Volume 1 (1951), pp. 5-31

[4] D'Aristotle, A. J. Quasicompactness and functionally Hausdorff spaces, J. Aust. Math. Soc., Volume 15 (1973), pp. 319-324

[5] Engelking, R. General Topology, revised and completed ed., Heldermann Verlag, Berlin, 1989

[6] Fox, R. H. On topologies for function spaces, Bull. Amer. Math. Soc., Volume 51 (1945), pp. 429-432

[7] Frolik, Z. Generalization of compact and Lindelöf spaces, Czechoslovak Math. J., Volume 13 (1959), pp. 172-217

[8] Gillman, L.; Jerison, M. Rings of continuous functions, Springer-Verlag, New York, 1960

[9] Gulick, D. The \(\sigma\)-compact-open topology and its relatives, Math. Scand., Volume 30 (1972), pp. 159-176

[10] N. C. Heldermann Developability and some new regularity axioms, Can. J. Math., Volume 33 (1981), pp. 641-663

[11] Hewitt, E. On two problems of Urysohn, Ann. of Math., Volume 47 (1946), pp. 503-509

[12] Jackson, J. R. Comparison of topologies on function spaces, Proc. Amer. Math. Soc., Volume 3 (1952), pp. 156-158

[13] Kohli, J. K.; Das, A. K. A class of spaces containing all generalized absolutely closed (almost compact) spaces, Appl. Gen. Topol., Volume 7 (2006), pp. 233-244

[14] Kohli, J. K.; Singh, D. Between compactness and quasicompactness, Acta Math. Hungar., Volume 106 (2005), pp. 317-329

[15] Kundu, S.; Garg, P. The pseudocompact-open topology on C(X), Topology Proc., Volume 30 (2006), pp. 279-299

[16] Kundu, S.; Raha, A. B. The bounded-open topology and its relatives, Rend. Istit. Mat. Univ. Trieste, Volume 27 (1995), pp. 61-77

[17] McArthur, W. G. \(G_\delta\)-diagonals and metrization theorems, Pacific J. Math., Volume 44 (1973), pp. 1-613

[18] McCoy, R. A. Second countable and separable function spaces, Amer. Math. Monthly, Volume 85 (1978), pp. 487-489

[19] McCoy, R. A.; Ntantu, I. Topological Properties of Spaces of Continuous Functions, Springer-Verlag, Berlin, 1988

[20] Osipov, A. V. The C-compact-open topology on function spaces, Topology Appl., Volume 159 (2012), pp. 3059-3066

[21] Pervin, W. J. Foundations of general topology, Academic Press, New York, 1964

[22] Steen, L. A.; Seebach, J. A. Counter Examples in Topology, Springer Verlag, New York, 1978

[23] Taylor, A. E.; Lay, D. C. Introduction to Functional Analysis, 2nd ed., John Wiley & Sons, New York, 1980