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arXiv · 1606.05445

A Few Notes on Formal Balls

Jean Goubault-Larrecq·Kok Min Ng
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Abstract

Using the notion of formal ball, we present a few 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 $\bar{\mathbb{R}}_+$-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. The point is that all those results reduce to domain-theoretic constructions on posets of formal balls.

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BibTeXRIS

Jean Goubault-Larrecq, Kok Min Ng. 2017-11-24. A Few Notes on Formal Balls. https://doi.org/10.23638/lmcs-13(4%3A18)2017

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