Pré-Publication, Document De Travail Année : 2016

A few notes on formal balls

Résumé

Using the notion of formal ball, we present a few easy, new results in the theory of quasi-metric spaces. With no specific order: every continuous Yoneda-complete quasi-metric space is sober and convergence Choquet-complete hence Baire in its d-Scott topology; for standard quasi-metric spaces, algebraicity is equivalent to having enough center points; on a standard quasi-metric space, every lower semicontinuous R+-valued function is the supremum of a chain of Lipschitz Yoneda-continuous maps; the continuous Yoneda-complete quasi-metric spaces are exactly the retracts of algebraic Yoneda-complete quasi-metric spaces; every continuous Yoneda-complete quasi-metric space has a so-called quasi-ideal model, generalizing a construction due to K. Martin; every continuous valuation on a continuous Yoneda-complete quasi-metric space extends to a Borel measure, and is a directed supremum of simple valuations. The point is that all those results reduce to domain-theoretic constructions on posets of formal balls.

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hal-01332749 , version 1 (16-06-2016)
hal-01332749 , version 2 (30-06-2016)

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Jean Goubault-Larrecq. A few notes on formal balls. 2016. ⟨hal-01332749v1⟩
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