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

机译:形式球的空间和拟度量空间的模型

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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, is contained in) of a quasi-metric space (X, d). Following Edalat and Heckmann, we prove that the order properties of (BX, is contained in) are tightly connected to topological properties of (X, d). In particular, we prove that (BX, is contained in) 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, is contained in, <). 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 contained in, <) 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.
机译:在本文中,我们使用域理论研究准度量空间。本文的主要目的是研究拟度量空间的最大点空间问题。在这里,我们证明可以通过连续dcpo来建模满足某些完整性属性(例如Yoneda和Smyth完整性)的准度量空间。为了实现此目标,我们首先研究准度量空间(X,d)的部分有序的形式球(包含在BX中)。继Edalat和Heckmann之后,我们证明了(BX,包含在)中的有序性质与(X,d)的拓扑性质紧密相关。特别是,如果(X,d)是代数Yoneda完全的,我们证明(BX,被包含在)是一个连续的dcpo。此外,我们证明了这种构造为Smyth完全拟度量空间提供了一个模型。然后,对于给定的准度量空间(X,d),我们引入部分有序的抽象形式球集{BX,包含在<)中。我们证明,如果拟度量空间(X,d)的共轭空间(X,d〜(-1))是正确的K完全,则(BX,包含在,<)中的理想完成是(X,d)的模型。此构造为任何Yoneda完全准度量空间(X,d)以及Sorgenfrey线,Kofner平面和Michael线提供了模型。

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  • 来源
    《Mathematical structures in computer science》 |2009年第2期|337-355|共19页
  • 作者单位

    School of Mathematics and Computer Science, Amirkabir University of Technology, 424 Hafez Ave.,Tehran, Iran, 15914;

    Faculty of Mathematics and Computer Sceince, Shahid Bahonar University,22 Bahman Blvd., Kerman, Iran, 76169-14111;

    School of Mathematics and Computer Science, Amirkabir University of Technology, 424 Hafez Ave.,Tehran, Iran, 15914;

    School of Mathematics and Computer Science, Amirkabir University of Technology, 424 Hafez Ave.,Tehran, Iran, 15914;

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