Zero dimensional and connected domains

We study continuous domains with bottomD such that every principal ideal inD is compact in the Lawson topology ofD. This class contains all objects of cartesian closed categories of continuous domains with bottom; it is also closed under arbitrary products and projection-embedding pairs. We employ the notion of apospace and useKoch's Arc Theorem to show that such a domainD is zero dimensional in its Lawson topology iffD is algebraic. This also classifies those continuous domainsD with bottom for which the Lawson topology λ(D) is a Stone Space. Finally, we give a criterion for the connectedness of the space λ(D) in terms of the poset of finite elements ofD and in terms of arc chains connecting bottom with every other element.