Convex power domain and vietoris space

Abstract

In this paper, the maximal point spaces (MP-space in short) of convex power domains are investigated. Some characterizations of the maximal points of convex power domains are obtained. It is proved that for a Scott compact continuous domain D, convex power domain C(D) is a domain hull of its maximal points Max( C(D)) if each element of Max( C(D)) is generated by a compact subset of Max( D). In this case, the space Max(C(D)) can be identified with the compact subsets Com(Max( D)) of Max( D) and the Vietoris topology on Com(Max( D)) is the topology inherited from the convex power domain. Finally, an example is given to show that even for a weakly compact continuous domain, its convex power domain need not be a domain hull of the maximal points.