Maximal point spaces of dcpos satisfying the Lawson condition

A directed complete poset (dcpo, for short) is said to satisfy the Lawson condition if the restriction of the Scott topology and the Lawson topology on the set of maximal points coincide. In this paper we investigate such dcpos, in particular their maximal point spaces. The main result is that for any dcpo satisfying the Lawson condition, the maximal point space is well-filtered and coherent (the intersection of any two saturated compact sets is compact). The relationship of such dcpos with other classes of dcpos are also considered.