The space of formal balls and models of quasi-metric spaces

In this paper we study quasi-metric spaces using domain theory. Our main objective in this paper is to study the maximal point space problem for quasi-metric spaces. Here we prove that quasi-metric spaces that satisfy certain completeness properties, such as Yoneda and Smyth completeness, can be modelled by continuous dcpo's. To achieve this goal, we first study the partially ordered set of formal balls (BX, subset of) of a quasi-metric space (X, d). Following Edalat and Heckmann, we prove that the order properties of (BX, subset of) are tightly connected to topological properties of (X, d). In particular, we prove that (BX, subset of) is a continuous dcpo if (X,d) is algebraic Yoneda complete. Furthermore, we show that this construction gives a model for Smyth-complete quasi-metric spaces. Then, for a given quasi-metric space (X,d), we introduce the partially ordered set of abstract formal balls (BX, subset of, (sic)). We prove that if the conjugate space (X, d(-1)) of a quasi-metric space (X, d) is right K-complete, then the ideal completion of (BX, subset of, (sic)) is a model for (X, d). This construction provides a model for any Yoneda-complete quasi-metric space (X,d), as well as the Sorgenfrey line, Kofner plane and Michael line.