External Characterizations of Continuous sL-domains

Abstract

In this paper a dcpo L is defined to be an sL-domain if all the principal ideals of L are join semilattices. With this concept, it is proved that a dcpo L is a continuous sL-domain if and only if [X → L] is a continuous dcpo (resp: sL-domain) for all compact and core compact spaces X. This characterization solves Problem 532 posed by J. D. Lawson and M. W. Mislove in the book [J. Van Mill, G. M. Reed (Editors), Open Problems in Topology, Elsevier Sci. Publishers B.V., North-Holland, 1990, pp 351–372]. Via idl (L), some other external characterizations of continuous sL-domains are also given.