Scott approach distance on metric spaces

TL;DR

This paper introduces the Scott distance in metric spaces, exploring its properties and relationships to approach spaces, and characterizes injective T0 approach spaces as cocomplete metric spaces with Scott distance.

Contribution

It defines the Scott distance in metric spaces and analyzes its properties, linking it to approach space theory and injective T0 spaces, providing new insights into their structure.

Findings

Scott distance makes metric spaces into approach spaces

Topological coreflection of Scott distance is between two Scott topologies

Injective T0 approach spaces are cocomplete and continuous metric spaces with Scott distance

Abstract

The notion of Scott distance between points and subsets in a metric space, a metric analogy of the Scott topology on an ordered set, is introduced, making a metric space into an approach space. Basic properties of Scott distance are investigated, including its topological coreflection and its relation to injective $T_0$ approach spaces. It is proved that the topological coreflection of the Scott distance is sandwiched between the $d$-Scott topology and the generalized Scott topology; and that every injective $T_0$ approach space is a cocomplete and continuous metric space equipped with its Scott distance.