Domain-Representability of Certain Complete Spaces

In this paper we show that three major classes of topological spaces are domain-representable, i.e., homeomorphic to the space of maximal elements of some domain (=continuous dcpo) with the relative Scott topology. The three classes are: T3 subcompact spaces, strongly α-favorable spaces with a Gδ-diagonal or with a base of countable order, and complete quasi-developable T3-spaces. It follows that any regular space with a monotonically complete base of countable order (in the sense of Wicke and Worrell) is domain-representable, as is any space with exactly one limit point. (In fact, any space with exactly one limit point is domain representable using a Scott domain.) The result on strongly α-favorable spaces with a Gδ-diagonal can be used to show that spaces such as the Sorgenfrey line, the Michael line, the Moore plane, the Nagata plane, and Heath’s V-space are domain-representable, and to show that a domain-representable space can be Hausdorff but not regular. MR Classifications: primary = 54E52; secondary = 54D70,54E30,06B35, 06F30