The local triangle axiom in topology and domain theory

Pawel Waszkiewicz

Abstract

We introduce a general notion of distance in weakly separated topological spaces. Our approach differs from existing ones since we do not assume the reflexivity axiom in general. We demonstrate that our partial semimetric spaces provide a common generalization of semimetrics known from Topology and both partial metrics and measurements studied in Quantitative Domain Theory. In the paper, we focus on the local triangle axiom, which is a substitute for the triangle inequality in our distance spaces. We use it to prove a counterpart of the famous Archangelskij Metrization Theorem in the more general context of partial semimetric spaces. Finally, we consider the framework of algebraic domains and employ Lebesgue measurements to obtain a complete characterization of partial metrizability of the Scott topology.

Keywords

Partial semimetric; Partial metric; Measurement; Lebesgue measurement; Local triangle axiom; Continuous poset; Algebraic dcpo

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1. A generalized contraction principle with control functions on partial metric spaces
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Computers & Mathematics with Applications  vol: 63  issue: 3  first page: 716  year: 2012  
doi: 10.1016/j.camwa.2011.11.035



Esta revista se publica bajo una licencia de Creative Commons Reconocimiento-NoComercial-SinObraDerivada 4.0 Internacional.

Universitat Politècnica de València

e-ISSN: 1989-4147   https://doi.org/10.4995/agt