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The topology of a quantale valued metric space

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The ‘the’ in the title hides a subtlety. A metric space induces not one but four topologies -by means of open sets, closed sets, closure, and interior -they just so happen to coincide. The agreement between these four structures arising from a metric function d : X × X → [0, ∞]is due to a combination of the metric axioms and the lattice structure of [0, ∞]. Further motivation materializes from Lawvere’s observation from 1973 to the effect that a (slightly generalized) metric space is a category enriched in [0, ∞]. Metric spaces taking values in structures other than [0, ∞]are relevant for generalizations of metric spaces and find a natural home in Lawvere’s categorical setting. In particular, in recent years quantales emerged as structures occupying an important niche in between [0, ∞]and arbitrary monoidal categories. Since a category enriched in a quantale Q is the same thing as a metric space taking values in one may ask whether such a thing belongs to algebra or geometry. Further, does the quadruplet of topologies associated to a Q-valued space/category still consist of identical siblings? We propose a litmus test for the geometricity of Q-valued spaces as we investigate these issues.
Original languageEnglish
Article number0
Pages (from-to)42-57
Number of pages16
JournalFuzzy Sets and Systems
Volume406
Early online date15 Jun 2020
DOIs
Publication statusPublished - 28 Feb 2021

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  • The topology of a quantale valued metric space

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