Set-open topologies on function spaces

Authors

DOI:

https://doi.org/10.4995/agt.2018.7630

Keywords:

set-open topology, pseudocompact-open topology, C-compact- open topology, quasi-uniform convergence topology, right K- completeness, α-continuous function

Abstract

Let X and Y be topological spaces, F(X,Y) the set of all functions from X into Y and C(X,Y) the set of all continuous functions in F(X,Y). We study various set-open topologies tλ (λ ⊆ P(X)) on F(X,Y) and consider their existence, comparison and coincidence in the setting of Y a general topological space as well as for Y = R. Further, we consider the parallel notion of quasi-uniform convergence topologies Uλ (λ ⊆ P(X)) on F(X,Y) to discuss Uλ-closedness and right Uλ-K-completeness properties of a certain subspace of F(X,Y) in the case of Y a locally symmetric quasi-uniform space. We include some counter-examples to justify our comments.

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Author Biographies

Wafa Khalaf Alqurashi, Umm Al-Qura University

Department of Mathematical Science, College of Applied Sciences

Liaqat Ali Khan, Quaid-i-Azam University

Department of Mathematics

Alexander V. Osipov, Ural State University of Economics

Krasovskii Institute of Mathematics and Mechanics, Ural Federal University

References

W. K. Alqurashi and L. A. Khan, Quasi-uniform convergence topologies on function spaces- Revisited, Appl. Gen. Top. 18, no. 2, (2017), 301-316. https://doi.org/10.4995/agt.2017.7048

R. F. Arens, A topology for spaces of transformations, Ann. Math. 47, no. 3 (1946), 480-495. https://doi.org/10.2307/1969087

R. Arens and J. Dugundji, Topologies for function spaces, Pacific J. Math. 1 (1951), 5-31. https://doi.org/10.2140/pjm.1951.1.5

A. Bouchair and S. Kelaiaia, Comparison of some set open topologies on C(X,Y), Topology Appl. 178, (2014), 352-359. https://doi.org/10.1016/j.topol.2014.10.008

A. Di Concilio and S. A. Naimpally, Some proximal set-open topologies, Boll. Unione Mat. Ital. (8) 1-B, (2000), 173-191.

P. Fletcher and W. F. Lindgren, Quasi-uniform spaces, Lecture Notes in Pure and Applied Mathematics, 77, Marcel Dekker, Inc., 1982.

R. Fox, On topologies for function spaces, Bull. Amer. Math. Soc. 51 (1945), 429-432. https://doi.org/10.1090/S0002-9904-1945-08370-0

D. Gale, Compact sets of functions and function rings, Proc. Amer. Math. Soc. 1 (1950), 303-308. https://doi.org/10.1090/S0002-9939-1950-0036503-X

D. Gulick, The σ-compact-open topology and its relatives, Math. Scand. 30 (1972), 159-176. https://doi.org/10.7146/math.scand.a-11072

D. Gulick and J. Schmets, Separability and semi-norm separability for spaces of bounded continuous functions, Bull. Soc. Roy. Sci. Leige 41 (1972), 254-260.

H. B. Hoyle, III, Function spaces for somewhat continuous functions, Czechoslovak Math. J. 21 (1971), 31-34.

J. R. Jackson, Comparison of topologies on function spaces, Proc. Amer. Math. Soc. 3 (1952), 156-158. https://doi.org/10.1090/S0002-9939-1952-0046031-5

J. L. Kelley, General topology, D. Van Nostrand Company, New York, 1955.

J. L. Kelley and I. Namioka, Linear topological spaces, D. Van Nostrand, 1963. https://doi.org/10.1007/978-3-662-41914-4

L. A. Khan and K. Rowlands, The σ-compact-open topology and its relatives on a space of vector-valued functions, Boll. Unione Mat. Italiana (7) 5-B, (1991), 727-739.

J. L. Kohli and J. Aggarwal, Closedness of certain classes of functions in the topology of uniform convergence, Demonstratio Math. 45 (2012), 947-952. https://doi.org/10.1515/dema-2013-0413

S. Kundu and R. A. McCoy, Topologies between compact and uniform convergence on function spaces, Internat. J. Math. Math. Sci. 16, no. 1 (1993), 101-110. https://doi.org/10.1155/S0161171293000122

S. Kundu and P. Garg, The pseudocompact-open topology on C(X), Topology Proceedings. Vol.~30, (2006), 279-299.

H.-P. A. Künzi, An introduction to quasi-uniform spaces, in: Beyond topology, Contemp. Math., 486, Amer. Math. Soc., Providence, RI, 2009, pp. 239-304. https://doi.org/10.1090/conm/486/09511

H.-P. A. Künzi and S. Romaguera, Spaces of continuous functions and quasi-uniform convergence, Acta Math. Hungar. 75 (1997), 287-298. https://doi.org/10.1023/A:1006593505036

A. S. Mashhour, I. A. Hasanein and S. N. El-Deeb, $alpha$-continuous and $alpha $-open mappings, Acta Math. Hungar. 41, (1983), 213-218. https://doi.org/10.1007/BF01961309

R. A. McCoy and I. Ntantu, Completeness properties of function spaces, Topology Appl. 22 (1986), 191-206. https://doi.org/10.1016/0166-8641(86)90009-X

R. A. McCoy and I. Ntantu, Topological properties of function spaces, Lecture Notes in Math. No. 1315, Springer-Verlag, 1988.

S. B. Myers, Equicontinuous sets of mappings, Ann. Math. 47 (1946), 496-502. https://doi.org/10.2307/1969088

S. A. Naimpally, Function spaces of quasi-uniform spaces, Indag. Math. 27 (1966), 768-771.

O. Njastad, On some classes of nearly open sets, Pacific J. Math. 15 (1965), 961-970. https://doi.org/10.2140/pjm.1965.15.961

S. E. Nokhrin, Some properties of set-open topologies, J. Math. Sci. 144 (2007), 4123-4151. https://doi.org/10.1007/s10958-007-0258-3

S. E. Nokhrin and A. V. Osipov, On the coincidence of set-open and uniform topologies, Proc. Steklov Inst. Math. Suppl. 267 (2009), 184-191. https://doi.org/10.1134/S0081543809070165

A. V. Osipov, The set-open topology, Topology Proc. 37 (2011), 205-217.

A. V. Osipov, The C-compact-open topology on function spaces, Topology Appl. 159, no. 13 (2012), 3059-3066. https://doi.org/10.1016/j.topol.2012.05.018

A. V. Osipov, Topological-algebraic properties of function spaces with set-open topologies, TTopology Appl. 159, no. 13 (2012), 800-805. https://doi.org/10.1016/j.topol.2011.11.049

A. V. Osipov, On the completeness properties of the C-compact-open topology on C(X), Ural Mathematical Journal 1, no. 1 (2015), 61-67. https://doi.org/10.15826/umj.2015.1.006

A. V. Osipov, Uniformity of uniform convergence on the family of sets, Topology Proc. 50 (2017), 79-86.

B. Papadopoulos, (Quasi) Uniformities on the set of bounded maps, Internat. J. Math. & Math. Scl. 17 (1994), 693-696. https://doi.org/10.1155/S0161171294000980

W. J. Pervin, Quasi-uniformization of topological spaces, Math. Ann. 147 (1962), 316-317. https://doi.org/10.1007/BF01440953

S. Romaguera, On hereditary precompactness and completeness in quasi-uniform spaces, Acta Math. Hungar. 73 (1996), 159-178. https://doi.org/10.1007/BF00058951

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Published

2018-04-02

How to Cite

[1]
W. K. Alqurashi, L. A. Khan, and A. V. Osipov, “Set-open topologies on function spaces”, Appl. Gen. Topol., vol. 19, no. 1, pp. 55–64, Apr. 2018.

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