On the greatest splitting topology

Abstract

It is well known that (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, in: Elliott Pearl (Ed.), Function Space Topologies, Open Problems in Topology, vol. 2, Elsevier, 2007, pp. 15–22]) 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. However, this intersection maybe not admissible. In the case, where Y is a locally compact Hausdorff space the compact-open topology on the set C ( Y , Z ) is splitting and admissible (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]), which means that the intersection of all admissible topologies on C ( Y , Z ) is admissible. In [R. Arens, J. Dugundji, Topologies for function spaces, Pacific J. Math. 1 (1951) 5–31] an example of a non-locally compact Hausdorff space Y is given having the same property for the case, where Z = [ 0 , 1 ] , that is on the set C ( Y , [ 0 , 1 ] ) the compact-open topology is splitting and admissible. This space Y is the set [ 0 , 1 ] with a topology τ, whose semi-regular reduction coincides with the usual topology on [ 0 , 1 ] . Also, in [R. Arens, J. Dugundji, Topologies for function spaces, Pacific J. Math. 1 (1951) 5–31, Theorem 5.3] another example of a non-locally compact space Y is given such that the compact-open topology on the set C ( Y , [ 0 , 1 ] ) is distinct from the greatest splitting topology. In this paper first we construct non-locally compact Hausdorff spaces Y such that the intersection of all admissible topologies on the set C ( Y , Z ) , where Z is an arbitrary regular space, is admissible. Furthermore, for a Hausdorff splitting topology t on C ( Y , Z ) we find sufficient conditions in order that t to be distinct from the greatest splitting topology. Using this result, we construct some concrete non-locally compact spaces Y such that the compact-open topology on C ( Y , Z ) , where Z is a Hausdorff space, is distinct from the greatest splitting topology. Finally, we give some open problems.