Borel structures for function spaces

If X and Y are topological spaces, then yX denotes the set of all continuous mappings from X into Y. For a given topology on yX, we may ask whether yX the natural mapping >< X --+ Y defined by (f, x) f(x) is continuous; if it is, then the topology on yX is said to be admissible [1]. It is always possible to find an admissible topology; for instance, the discrete topology on YX is always admissible. Moreover, when X is locally compact, YX has a unique smallest admissible topology; this is the familiar "compactopen" topology. These and related questions concerning topologies for function spaces have been investigated in considerable detail by several authors [1, 4]. We are interested in the analogous situation when X and Y are Borel spaces rather than topological spaces; in this case we define YX as the set of all Borel mappings from X into Y. Unfortunately, it turns out that even for some of the simplest Borel spaces, it is impossible to define a Borel structure on YX so that is a Borel mapping; even if we impose the discrete structure on YX, will in general not be Borel. As a substitute, we may ask ourselves the following questions" For which subsets F of Y is it possible to impose a Borel structure on F so that IF >< X will be Borel? If is is possible for a given F, what can we say about the appropriate structures? In particular, is there always a smallest such structure (corresponding to the compact-open topology) ? Let us introduce some terminology. We will write "space" instead of "Borel space", "structure" instead of "Borel structure", and F instead of F >< X. A structure R on F for which F is Borel will be called admissible; a subset F of yX on which it is possible to impose an admissible structure is