A Few Notes on Formal Balls

TL;DR

This paper explores properties of quasi-metric spaces using formal balls, establishing new results on their topological and domain-theoretic characteristics, including sobriety, algebraicity, and models, with implications for their structure and functions.

Contribution

It introduces novel results linking formal ball constructions to the topological and domain-theoretic properties of quasi-metric spaces, including sobriety, algebraicity, and models.

Findings

Continuous Yoneda-complete spaces are sober and Baire.

Algebraicity is equivalent to having enough center points in standard spaces.

Lower semicontinuous functions are supremums of Lipschitz Yoneda-continuous maps.

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.