A study of function space topologies for multifunctions


  • Ankit Gupta University of Delhi
  • Ratna Dev Sarma University of Delhi




multifunction, topology, function space, continuous convergence, splittingness, admissibility


Function space topologies are investigated for the class of continuous multifunctions. Using the notion of continuous convergence, splittingness and admissibility are discussed for the topologies on continuous multifunctions. The theory of net of sets is further developed for this purpose. The (τ,μ)-topology on the class of continuous multifunctions is found to be upper admissible, while the compact-open topology is upper splitting. The point-open topology is the coarsest topology which is coordinately admissible, it is also the finest topology which is coordinately splitting. 


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Author Biographies

Ankit Gupta, University of Delhi

Department of Mathematics

Ratna Dev Sarma, University of Delhi

Department of Mathematics, Rajdhani College


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

J. Cao, I. L. Reilly and M. V. Vamanamurthy, Comparison of convergences for multifunctions, Demonstratio Math. 30 (1997), 171-182.

S. Dolecki and F. Mynard, A unified theory of function spaces and hyperspaces: local properties, Houston J. Math. 40, no. 1 (2014), 285-318.

D. N. Georgiou and S. D. Iliadis, On the greatest splitting topology, Topology Appl. 156 (2008), 70-75. https://doi.org/10.1016/j.topol.2007.11.008

D. N. Georgiou, S. D. Iliadis and B. K. Papadopoulos, Topology on function spaces and the coordinate continuity, Topology Proc. 25 (2000), 507-517.

D. N. Georgiou, S. D. Iliadis and B. K. Papadopoulos, On dual topologies, Topology Appl. 140 (2004), 57-68. https://doi.org/10.1016/j.topol.2003.08.015

A. Gupta and R. D. Sarma, Function space topologies for generalized topological spaces, J. Adv. Res. Pure Math. 7, no. 4 (2015), 103-112.

A. Gupta and R. D. Sarma, On dual topologies concerning function spaces over $mathcal{C}_{mu, nu}(Y,Z)$, preprint.

V. G. Gupta, Compact convergence for fultifunctions, Pure Appl. Math. Sci. 17 (1983), 35-40.

V. G. Gupta, Compact convergence topology for multi-valued functions, Proc. Nat. Acad. Sci. India Sect. A 53 (1983), 164-167.

S. Hu and N. S. Papageorgiou, Handbook of multivalued analysis, Vol. I Theory, Kluwer Academic Publishers, Dordrecht, 1997.

P. Jain and S. P. Arya, Some function space topologies for multifunctions, India J. Pure Appl. Math. 6 (1975), 1488-1506.

F. Jordan, Coincidence of function space topologies, Topology Appl. 157 (2010), 336-351. https://doi.org/10.1016/j.topol.2009.09.002

E. Klein and A. Thompson, Theory of correspondences: including applications to mathematical economics, Canadian Mathematical Society Series of Monographs and Advanced texts. J. Wiley & Sons, 1984.

V. J. Mancuso, An Ascoli theorem for multi-valued functions, J. Austral. Math. Soc. 12 (1971), 466-472. https://doi.org/10.1017/S1446788700010351

S. Mrowka, On Convergence of nets of sets, Fund. Math. 45 (1958) 237-246.

K. Porter, The open-open topology for function spaces, Inter. J. Math. and Math. Sci. 12 (1993), 111-116. https://doi.org/10.1155/S0161171293000134

M. Przemski, On continuous convergence for nets of multifunctions, Demonstratio Math. 44 (2011), 181-200. https://doi.org/10.1515/dema-2013-0292

R. D. Sarma, On convergence in generalized topology, Int. J. Pure Appl. Math. 60, no. 2 (2010), 205-210.

R. E. Smithson, Topologies on sets of relations, J. Natur. Sci. and Math. 11 (1971), 43-50.

R. E. Smithson, Uniform convergence for multifunctions, Pacific J. Math. 39 (1971), 253-259. https://doi.org/10.2140/pjm.1971.39.253

R. E. Smithson, Multifunctions, Nieuw. Arch. Wisk. 20, no. 3 (1972), 31-53.

L. A. Steen and J. A. Seebach, Counterexamples in topology, Springer, New York 1978. https://doi.org/10.1007/978-1-4612-6290-9




How to Cite

A. Gupta and R. D. Sarma, “A study of function space topologies for multifunctions”, Appl. Gen. Topol., vol. 18, no. 2, pp. 331–344, Oct. 2017.