Quasi-uniform convergence topologies on function spaces- Revisited

Wafa Khalaf Alqurash, Liaqat Ali Khan


Let X and Y be topological space and F(X,Y) the set of all functions from X into Y. We study various quasi-uniform convergence topologies U_{A} (A⊆P(X)) on F(X,Y) and their comparison in the setting of Y a quasi-uniform space. Further, we study U_{A}-closedness and right K-completeness properties of certain subspaces of generalized continuous functions in F(X,Y) in the case of Y a locally symmetric quasi-uniform space or a locally uniform space.


Quasi-uniform space, topology of quasi-uniform convergence on a family of sets, locally uniform spaces, right K-completeness, quasi-continuous functions, somewhat continuous functions.

Subject classification

54C35, 54E15, 54C08

Full Text:



A. Andrikopoulos, Completeness in quasi-uniform spaces, Acta Math. Hungar. 105 (2004), 151-173.


R. F. Arens, A topology for spaces of transformations, Ann. of Math. 47, no. 2 (1946), 480-495.


R. Arens and J. Dugundji, Topologies for function spaces, Pacific J. Math. 1 (1951), 5-31.


A. Bouchair and S. Kelaiaia, Comparison of some set open topologies on C(X,Y), Topology Appl. 178 (2014), 352-359.


R. C. Buck, Bounded continuous functions on a locally compact space, Michigan Math. J. 5 (1958), 95-104.


J. Cao, Answers to two questions of Papadopoulos, Questions Answers Gen. Topology 14 (1996), 111-116.

S. Cobzas, Functional Analysis in Asymmetric Normed Spaces, Springer, 2013.

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

D. Doitchinov, A concept of completeness of quasi-uniform spaces, Topology Appl. 38 (1991), 205-217.


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.


K. R. Gentry, H. B. Hoyle, III, Somewhat continuous functions, Czechoslovak Math. J. 21 (1971), 5-12.

D. Gulick, The $sigma $-compact-open topology and its relatives, Math. Scand. 30 (1972) 159-176.


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

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

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

J. L. Kelley, I. Namioka and Co-authors, Linear Topological Spaces, D. Van Nostrand, 1963.


L. A. Khan, Some approximation results for the compact-open topology, Periodica Math. Hungar. 30 (1995), 81-86.


L. A. Khan, Linear Topological Spaces of Continuous Vector-valued Functions, Academic Publications, 2013 https://doi.org/0.12732/acadpubl.201301

L. A. Khan and K. Rowlands, The $sigma $-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. 1. (2012), 947-952.

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.


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


H. P. A. Künzi and S. Romaguera, Completeness of the quasi-uniformity of quasi-uniform convergence, Papers on general topology and applications (Gorham, ME, 1995), 231-237, Ann. New York Acad. Sci., 806, New York Acad. Sci., New York, 1996.

H. P. A. Künzi and S. Romaguera, Spaces of continuous functions and quasi-uniform convergence, Acta Math. Hungar. 75 (1997), 287-298.


P. Th. Lambrinos, On precompact (quasi-) uniform structures, Proc. Amer. Math. Soc. 62 (1977), 365-366.


R. A. McCoy and I. Ntantu, Topological Properties of Function Spaces, Lectures Notes in Math. No. 1315, Springer-Verlag, 1988.

M. G. Murdeshwar, and S. A. Naimpally, Quasi-uniform Topological Spaces, Noordhoff Groningen, 1966.

M. G. Murdeshwar and K. K. Theckedath, Boundedness in a quasi-uniform space, Canad. Math. Bull. 13 (1970), 367-370.


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

S. A. Naimpally, On strongly continuous functions, Amer. Math. Monthly 74 (1967), 166-169.


S. E. Nokhrin, Some properties of set-open topologies, J. Math. Sci. 144 (2007), 4123-4151.


S. E. Nokhrin and A.V. Osipov, On the coincidence of set-open and uniform topologies, Proc. Steklov Inst. Math. 267, Suppl. 3 (2009), 184-191.


A. V. Osipov, The set-open topology, Topology Proc. 37 (2010), 205-217.

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. Sci. 17 (1994), 693-696.


B. Papadopoulos, A note on the paper "Quasi-uniform convergence on function spaces", Q & A in General Topology 13 (1995), 55-56.

W. J. Pervin, Quasi-uniformization of topological spaces, Math. Ann. 147 (1962), 316-317.


M. Racila, On some properties of quasi-uniform convergence and quasi-continuous functions, Real Anal. Exchange 24 (1998/99), 791-797.

S. Romaguera, On hereditary precompactness and completeness in quasi-uniform spaces, Acta Math. Hungar. 73 (1996), 159-178.


J. L. Sieber and W.J. Pervin, Completeness in quasi-uniform spaces, Math. Ann. 158 (1965), 79-81.


M. B. Smyth, Completeness of quasi-uniform and syntopological spaces, J. London Math. Soc. 49, no. 2 (1994), 385-400.


R. Stoltenberg, Some properties of quasi-uniform spaces, Proc. London Math. Soc. 17, no. 3 (1967), 226-240.


Ph. Sünderhauf, Quasi-uniform completeness in terms of Cauchy nets, Acta Math. Hungar. 69 (1995), 47-54.


J. Williams, Locally uniform spaces, Trans. Amer. Math. Soc. 168 (1972), 435-469.


Abstract Views

Metrics Loading ...

Metrics powered by PLOS ALM

Esta revista se publica bajo una licencia de Creative Commons Reconocimiento-NoComercial-SinObraDerivada 4.0 Internacional.

Universitat Politècnica de València

e-ISSN: 1989-4147   https://doi.org/10.4995/agt