## Jean Goubault-Larrecq ; Kok Min Ng - A Few Notes on Formal Balls

lmcs:2021 - Logical Methods in Computer Science, November 28, 2017, Volume 13, Issue 4 - https://doi.org/10.23638/LMCS-13(4:18)2017
A Few Notes on Formal Balls

Authors: Jean Goubault-Larrecq ; Kok Min Ng

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.

Volume: Volume 13, Issue 4
Published on: November 28, 2017
Accepted on: November 28, 2017
Submitted on: September 6, 2016
Keywords: Mathematics - General Topology,54E99