In order to describe those spaces with a domain environment, some new lower axioms of separation are introduced and studied from the both of topology and domain theory respectively in the paper.

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 continuous domain D, convex power domain C(D) is a domain hull of its maximal points Max(C(D)) if and only 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 identi-ed 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.

In this paper it is proved that a space may be realized as the set of the maximal elements in a continuous poset if and only if it is Tychonoff.