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

Volume 9, Issue 6, pp 3511--3518 http://dx.doi.org/10.22436/jnsa.009.06.06

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Authors

Deniz Tokat - Department of Mathematics, Faculty of Arts and Sciences, Nevsehir Hacı Bektas Veli University, 50300 Nevsehir, Turkey. İsmail Osmanoğlu - Department of Mathematics, Faculty of Arts and Sciences, Nevsehir Hacı Bektas Veli University, 50300 Nevsehir, Turkey.

Abstract

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).

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Keywords

• Function space
• set-open topology
• compact-open topology
• quasicompactness
• separability
• submetrizability
• second countability.

MSC

•  54C35
•  54D65
•  54E35

References

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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


• [17] W. G. McArthur, \(G_\delta\)-diagonals and metrization theorems, Pacific J. Math., 44 (1973), 613{617


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


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


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


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


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


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