The regular spaces which may be realized as the set of maximal elements in an !-continuous dcpo are the Polish spaces. In addition, we give a new and conceptually simple model for complete metric spaces.

Ideal domains have an elementary order theoretic structure: Every element is either compact or maximal. Despite this, we establish that (1) They can model any space currently known to possess a countably based model, and (2) The metric spaces with ideal models are exactly the completely metrizable spaces.

We prove that a metric space may be realized as the set of maximal elements in a continuous dcpo if and only if it is completely metrizable by showing more generally that the space of maximal elements in a domain is always complete in a sense rst introduced by Choquet.