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.