Coincidence of the Isbell and Scott topologies on the function spaces of quasicontinuous domains

Abstract

The function space plays an indispensable role in investigating the Cartesian closedness of categories of domains. This paper focuses on the quasicontinuity and topological properties of the function spaces associated with quasicontinuous domains. For any quasicontinuous domain P with property M⁎ and bounded complete algebraic domain X, it is shown that: (i) the function space [X→P] is a quasicontinuous domain; especially, a counterexample is given to illustrate that property M⁎ is necessary for the quasicontinuity of the function space; (ii) the Isbell topology and Scott topology are equal in the function space [X→P].