Approaching metric domains

TL;DR

This paper characterizes a class of metric spaces called injective T0 approach spaces as continuous lattices with a specific algebraic structure, linking topology and metric space theory.

Contribution

It provides a precise characterization of injective T0 approach spaces as continuous lattices with a unital, associative $[0, olinebreak ext{infty}]$-action, extending the understanding of cocompleteness in approach spaces.

Findings

Injective T0 approach spaces are exactly continuous lattices with a $[0, ext{infty}]$-action.

A thorough analysis of cocompleteness in approach spaces was conducted.

The study bridges the gap between topology, metric spaces, and lattice theory.

Abstract

In analogy to the situation for continuous lattices which were introduced by Dana Scott as precisely the injective T$_0$ spaces via the (nowadays called) Scott topology, we study those metric spaces which correspond to injective T$_0$ approach spaces and characterise them as precisely the continuous lattices equipped with an unitary and associative $[0,\infty]$-action. This result is achieved by a thorough analysis of the notion of cocompleteness for approach spaces.