The local triangle axiom in topology and domain theory

Pawel Waszkiewicz

United Kingdom

University of Birmingham

School of Computer Science

Submitted: 2013-12-11

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Accepted: 2013-12-11

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Published: 2003-04-01

DOI: https://doi.org/10.4995/agt.2003.2009
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Keywords:

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

Supporting agencies:

This research was not funded

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.
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