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

Published online by Cambridge University Press:  01 April 2009

M. ALI-AKBARI ,
B. HONARI ,
M. POURMAHDIAN  and
M. M. REZAII
M. ALI-AKBARI
Affiliation:
School of Mathematics and Computer Science, Amirkabir University of Technology, 424 Hafez Ave., Tehran, Iran, 15914
B. HONARI
Affiliation:
Faculty of Mathematics and Computer Sceince, Shahid Bahonar University, 22 Bahman Blvd., Kerman, Iran, 76169-14111
M. POURMAHDIAN*
Affiliation:
School of Mathematics and Computer Science, Amirkabir University of Technology, 424 Hafez Ave., Tehran, Iran, 15914
M. M. REZAII
Affiliation:
School of Mathematics and Computer Science, Amirkabir University of Technology, 424 Hafez Ave., Tehran, Iran, 15914
*
§Corresponding author – Email: pourmahd@ipm.ir. This author was partially supported by the Institute for Studies in Theoretical Physics and Mathematics, Tehran, Iran, grant No. 85030112.
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Abstract

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, ⊑) of a quasi-metric space (X, d). Following Edalat and Heckmann, we prove that the order properties of (BX, ⊑) are tightly connected to topological properties of (X, d). In particular, we prove that (BX, ⊑) 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, ⊑, ≺). 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, ⊑, ≺) 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.

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 19 , Issue 2 , April 2009 , pp. 337 - 355
Copyright
Copyright © Cambridge University Press 2009

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