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|Paper IPM / M / 8970||
In this paper we study quasi-metric spaces via 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 which satisfy certain completeness properties such as Yoneda and Smyth completeness can be modeled by continuous dcpo's. To achieve this goal, we first study the partially ordered set of formal balls (BX, \sqsubseteq) of a quasi-metric space (X, d). Following Edalat and Heckmann, we prove that the order properties of (BX, \sqsubseteq) are tightly connected with topological properties of (X, d). In particular, we prove that (BX, \sqsubseteq) is a continuous dcpo if (X, d) is algebraic Yoneda-complete. Furthermore, we show that this construction does give 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, \sqsubseteq, \prec). We prove that if the conjugate space (X, d−1) of a quasi-metficspace (X, d) is right K-complete then the ideal completion of (BX, \sqsubseteq, \prec) is a model for (X, d). This construction provides a model for any Yoneda-complete quasi-metric space (X, d) as well as Sorgenfrey line, Kofner plane and Michaelline.
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