Topological and Metric Approach Spaces

Abstract

Both topological and metric spaces can be viewed as special types of approach spaces. More precisely, both the categories of topological spaces and continuous maps, and of (quasi)-metric spaces and nonexpansive maps, can be embedded as full and isomorphism-closed subcategories of the category of approach spaces, the former as a stable (i.e. simultaneously concretely reflective and concretely coreflective) subcategory and the latter as a concretely coreflective subcategory.

Well, I use the metric system. It’s the only way to get really exact numbers.

(Catherynne M. Valente, in The Girl Who Fell Beneath

Fairyland and Led the Revels There)

As every mathematician knows, nothing is more fruitful than these obscure analogies, these indistinct reflections of one theory into another, these furtive caresses, these inexplicable disagreements ...

(André Weil)