Set-open topologies on function spaces. (English) Zbl 1392.54020

Let \(X\) and \(Y\) be topological spaces and \(F(X,Y)\) (\(C(X,Y)\)) be the set of all (continuous) functions \(f:X\to Y\). The authors consider different types of topologies defined on \(F(X,Y)\). The first type consists of various set-open topologies defined as follows. Let \(\lambda\subset\mathcal{P}(X)\) be a network which covers \(X\). For any \(A\in\lambda\) and open \(G\subset Y\), let \(N(A,G) = \{f \in F(X,Y) : f(A)\subset G\}\). Then the collection \(\{N(A,G) : A\in\lambda, \text{ open } G \subseteq Y \}\) forms a subbase for a “set-open” topology on \(F(X, Y)\), called the \(\lambda\)-open topology and denoted \(t_\lambda\). In the first part of the paper under review the authors study existence, comparison and coincidence of such topologies in the setting of \(Y\) a general topological space as well as for \(Y = \mathbb{R}\). In addition to the \(t_\lambda\)-topology, the authors define the notion of \(t_{\lambda^\ast}\)-topology on \(C(X,Y)\) which has as a subbase the collection \(\{ N^\ast_c(A,G): A\in\lambda, \text{ open }G\subset Y\}\), where \(N^\ast_c(A,G)=\{ f\in C(X,Y): \overline{f(A)}\subset G\}\).
In the next parts of the paper the authors also consider the topology of uniform convergence on elements of \(\lambda\) on \(C(X,Y)\), denoted by \(C_{\lambda,u}(X,Y)\), which has as a base at each \(f\in C(X,Y)\) the collection \(\{ \langle f,A, \varepsilon\rangle: A\in\lambda, \varepsilon> 0\}\), where \(\{ \langle f,A, \varepsilon\rangle=\{ g\in C(X,Y): \operatorname{sup}_{x\in A}\varrho(f(x),g(x))<\varepsilon\}\).
Finally the authors consider the notions of quasi-uniform convergence topologies on \(F(X,Y)\), which are parallel to those of the set-open topologies. Let \((Y, \mathcal{U})\) be a quasi-uniform space, and let \(\lambda\subset\mathcal{P}(X)\) be a collection which covers \(X\). For any \(A\in\lambda\) and \(U\in\mathcal{U}\), let \(\hat{U}|A=\{ (f,g)\in F(X,Y)\times F(X,Y): (f(x),g(x))\in U \text{ for all }x\in A\}\). Then the collection \(\{ \hat{U}|A : A\in\lambda \text{ and }U\in \mathcal U\}\) forms a subbase for a quasi-uniformity, called the quasi-uniformity of quasi-uniform convergence on the sets in \(\lambda\) induced by \(\mathcal{U}\). The induced topology on \(F(X, Y)\) is called the topology of quasi-uniform convergence on the sets in \(\lambda\) and is denoted by \(\mathcal{U}_\lambda\). The authors establish some results on closedness and completeness of the space \(C^\alpha(X, Y)\) of all \(\alpha\)-continuous functions from \(X\) into \(Y\) in the \(\mathcal{U}_X\)-topology.


54C35 Function spaces in general topology
54C30 Real-valued functions in general topology
54E15 Uniform structures and generalizations
54C08 Weak and generalized continuity
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[1] 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 · Zbl 1382.54007
[2] R. F. Arens, A topology for spaces of transformations, Ann. Math. 47, no. 3 (1946), 480-495. https://doi.org/10.2307/1969087 · Zbl 0060.39704
[3] 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 · Zbl 0044.11801
[4] 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 · Zbl 1305.54027
[5] A. Di Concilio and S. A. Naimpally, Some proximal set-open topologies, Boll. Unione Mat. Ital. (8) 1-B, (2000), 173-191. · Zbl 0942.54012
[6] P. Fletcher and W. F. Lindgren, Quasi-uniform spaces, Lecture Notes in Pure and Applied Mathematics, 77, Marcel Dekker, Inc., 1982. · Zbl 0501.54018
[7] 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 · Zbl 0060.41202
[8] 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 · Zbl 0037.35501
[9] 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 · Zbl 0253.46045
[10] 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. · Zbl 0243.46037
[11] H. B. Hoyle, III, Function spaces for somewhat continuous functions, Czechoslovak Math. J. 21 (1971), 31-34. · Zbl 0212.55002
[12] 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 · Zbl 0046.11703
[13] J. L. Kelley, General topology, D. Van Nostrand Company, New York, 1955.
[14] J. L. Kelley and I. Namioka, Linear topological spaces, D. Van Nostrand, 1963. https://doi.org/10.1007/978-3-662-41914-4 · Zbl 0318.46001
[15] 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. · Zbl 0766.54012
[16] 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 · Zbl 1272.54017
[17] 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 · Zbl 0798.54019
[18] S. Kundu and P. Garg, The pseudocompact-open topology on C(X), Topology Proceedings. Vol. 30, (2006), 279-299. · Zbl 1126.54008
[19] 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 · Zbl 1193.54014
[20] 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 · Zbl 0921.54011
[21] 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 · Zbl 0534.54006
[22] 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 · Zbl 0621.54011
[23] R. A. McCoy and I. Ntantu, Topological properties of function spaces, Lecture Notes in Math. No. 1315, Springer-Verlag, 1988. · Zbl 0647.54001
[24] S. B. Myers, Equicontinuous sets of mappings, Ann. Math. 47 (1946), 496-502. https://doi.org/10.2307/1969088 · Zbl 0061.24305
[25] S. A. Naimpally, Function spaces of quasi-uniform spaces, Indag. Math. 27 (1966), 768-771. · Zbl 0134.41703
[26] 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 · Zbl 0137.41903
[27] 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 · Zbl 1151.54014
[28] 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 · Zbl 1235.54010
[29] A. V. Osipov, The set-open topology, Topology Proc. 37 (2011), 205-217. · Zbl 1227.54021
[31] 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 · Zbl 1250.54019
[32] 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 · Zbl 1243.54037
[33] 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 · Zbl 1396.54019
[34] A. V. Osipov, Uniformity of uniform convergence on the family of sets, Topology Proc. 50 (2017), 79-86. · Zbl 1384.54011
[35] 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 · Zbl 0812.54024
[36] W. J. Pervin, Quasi-uniformization of topological spaces, Math. Ann. 147 (1962), 316-317. https://doi.org/10.1007/BF01440953 · Zbl 0101.40501
[37] S. Romaguera, On hereditary precompactness and completeness in quasi-uniform spaces, Acta Math. Hungar. 73 (1996), 159-178. https://doi.org/10.1007/BF00058951 · Zbl 0924.54035
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