Topologies on function spaces and hyperspaces

[EN] Let Y and Z be two fixed topological spaces, O(Z) the family of all open subsets of Z, C(Y,Z) the set of all continuous maps from Y to Z, and OZ(Y ) the set {f−1(U) : f ϵ C(Y,Z) and U ϵ O(Z)}. In this paper, we give and study new topologies on the sets C(Y,Z) and OZ(Y ) calling (A,A0)-splitting and (A,A0)-admissible, where A and A0 families of spaces.Work Supported by the Caratheodory programme of the University of PatrasGeorgiou, D. (2009). Topologies on function spaces and hyperspaces. Applied General Topology. 10(1):159-171. doi:10.4995/agt.2009.1794.159171101Arens, R. F. (1946). A Topology for Spaces of Transformations. The Annals of Mathematics, 47(3), 480. doi:10.2307/1969087Arens, R., & Dugundji, J. (1951). Topologies for function spaces. Pacific Journal of Mathematics, 1(1), 5-31. doi:10.2140/pjm.1951.1.5J. Dugundji, Topology, Allyn and Bacon, Inc. Boston 1968.R. Engelking, General Topology, Warszawa 1977.Fox, R. H. (1945). On topologies for function spaces. Bulletin of the American Mathematical Society, 51(6), 429-433. doi:10.1090/s0002-9904-1945-08370-0Georgiou, D. N., Iliadis, S. D., & Papadopoulos, B. K. (2004). On dual topologies. Topology and its Applications, 140(1), 57-68. doi:10.1016/j.topol.2003.08.015Gierz, G., Hofmann, K. H., Keimel, K., Lawson, J. D., Mislove, M. W., & Scott, D. S. (1980). A Compendium of Continuous Lattices. doi:10.1007/978-3-642-67678-9P. Lambrinos and B. K. Papadopoulos, The (strong) Isbell topology and (weakly) continuous lattices, Continuous Lattices and Applications, Lecture Notes in pure and Appl. Math. No. 101, Marcel Dekker, New York 1984, 191–211.F. Schwarz and S. Weck, Scott topology, Isbell topology, and continuous convergence, Lecture Notes in Pure and Appl. Math. No.101, Marcel Dekker, New York 1984, 251-271