On the compact-open and admissible topologies

Abstract

It is known (see, for example, [H. Render, Nonstandard topology on function spaces with applications to hyperspaces, Trans. Amer. Math. Soc. 336 (1) (1993) 101–119; M. Escardo, J. Lawson, A. Simpson, Comparing cartesian closed categories of (core) compactly generated spaces, Topology Appl. 143 (2004) 105–145; D.N. Georgiou, S.D. Iliadis, F. Mynard, Function space topologies, in: Open Problems in Topology 2, Elsevier, 2007, pp. 15–23]) that the intersection of all admissible topologies on the set C ( Y , Z ) of all continuous maps of an arbitrary space Y into an arbitrary space Z, is always the greatest splitting topology (which in general is not admissible). The following, interesting in our opinion, problem is arised: when a given splitting topology (for example, the compact-open topology, the Isbell topology, and the greatest splitting topology) is the intersection of k admissible topologies, where k is a finite number. Of course, in this case this splitting topology will be the greatest splitting. In the case, where a given splitting topology is admissible the above number k is equal to one. For example, if Y is a locally compact Hausdorff space, then k = 1 for the compact-open topology (see [R.H. Fox, On topologies for function spaces, Bull. Amer. Math. Soc. 51 (1945) 429–432; R. Arens, A topology for spaces of transformations, Ann. of Math. 47 (1946) 480–495; R. Arens, J. Dugundji, Topologies for function spaces, Pacific J. Math. 1 (1951) 5–31]). Also, if Y is a corecompact space, then k = 1 for the Isbell topology (see [P. Lambrinos, B.K. Papadopoulos, The (strong) Isbell topology and (weakly) continuous lattices, in: Continuous Lattices and Applications, in: Lect. Notes Pure Appl. Math., vol. 101, Marcel Dekker, New York, 1984, pp. 191–211; F. Schwarz, S. Weck, Scott topology, Isbell topology, and continuous convergence, in: Lect. Notes Pure Appl. Math., vol. 101, Marcel Dekker, New York, 1984, pp. 251–271]). In [R. Arens, J. Dugundji, Topologies for function spaces, Pacific J. Math. 1 (1951) 5–31] a non-locally compact completely regular space Y is constructed such that the compact-open topology on C ( Y , S ) , where S is the Sierpinski space, coincides with the greatest splitting topology (which is not admissible). This fact is proved by the construction of two admissible topologies on C ( Y , S ) whose intersection is the compact-open topology, that is k = 2 . In the present paper improving the method of [R. Arens, J. Dugundji, Topologies for function spaces, Pacific J. Math. 1 (1951) 5–31] we construct some other non-locally compact spaces Y such that the compact-open topology on C ( Y , S ) is the intersection of two admissible topologies. Also, we give some concrete problems concerning the above arised general problem.