A function space characterization of RB-domains

Abstract

We prove that a domain $D$ is an RB-domain if and only if the function space $[D\to E]$ is continuous for every domain $E$, thereby resolving a conjecture of Luan and Li. More precisely, for each domain $D$ we construct an explicit algebraic test domain $A_D$ such that continuity of the single function space $[D\to A_D]$ already forces $D$ to be an RB-domain. For countably based domains $D$, the test domain can be chosen as a fixed domain that is independent of $D$. The analogous continuity characterization of FS-domains is false: the closed-disk domain is an FS-domain, but its function space into a suitable pointed algebraic domain is not continuous.