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Paper IPM / M / 8970  


Abstract:  
In this paper we study quasimetric spaces via domain theory. Our main objective in this paper is to study the maximal point space problem for quasimetric spaces. Here we prove that quasimetric 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 quasimetric 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 Yonedacomplete. Furthermore, we show that this construction does give a model for Smythcomplete quasimetric spaces. Then, for a given quasimetric 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 quasimetficspace (X, d) is right Kcomplete then the ideal completion of (BX, \sqsubseteq, \prec) is a model for (X, d). This construction provides a model for any Yonedacomplete quasimetric space (X, d) as well as Sorgenfrey line, Kofner plane and Michaelline.
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