Ideal models of spaces. (Q1427784)

This paper is a continuation of the author's programme of studying when spaces can be `modelled' by representing them as the subspaces of maximal points of continuous posets equipped with the Scott topology. In the present paper he considers `ideal dcpo's' in which every element is either compact or maximal. The main result asserts that if \(D\) is an ideal domain with \(\max(D)\) metrizable, then \(\max(D)\) is a \(G_{\delta}\) subset of \(D\). As a consequence he obtains a remarkable characterization of completely metrizable spaces, as those metrizable spaces which have ideal models.