Completion of continuity spaces with uniformly vanishing asymmetry

TL;DR

This paper introduces a new completion method for continuity spaces with a property called uniformly vanishing asymmetry, extending classical completions of metric and uniform spaces to a broader context.

Contribution

It generalizes the filter completion to continuity spaces with uniformly vanishing asymmetry, broadening the scope of completion techniques beyond symmetric spaces.

Findings

Classical completions rely on symmetry, but this work extends to spaces with asymmetry.

The new completion method applies to continuity spaces with uniformly vanishing asymmetry.

This approach unifies and generalizes existing completion theories.

Abstract

The classical Cauchy completion of a metric space (by means of Cauchy sequences) as well as the completion of a uniform space (by means of Cauchy filters) are well-known to rely on the symmetry of the metric space or uniform space in question. For qausi-metric spaces and quasi-uniform spaces various non-equivalent completions exist, often defined on a certain subcategory of spaces that satisfy a key property required for the particular completion to exist. The classical filter completion of a uniform space can be adapted to yield a filter completion of a metric space. We show that this completion by filters generalizes to continuity spaces that satisfy a form of symmetry which we call uniformly vanishing asymmetry.