Uniformites et Continuity Spaces

TL;DR

This paper explores the structure of continuity spaces and their relation to quasi-uniform spaces, providing a framework for understanding uniformities through filterbases and metric-like functions.

Contribution

It introduces a formal definition of continuity spaces and demonstrates how they induce quasi-uniform structures via filterbases, extending the theory of uniform spaces.

Findings

Continuity spaces are equipped with a metric-like function d satisfying specific properties.

A set of positive elements in an abelian semigroup forms a basis for quasi-uniform filterbases.

Symmetrization of filterbases yields uniform filterbases, linking continuity spaces to uniform spaces.

Abstract

A semigroup A is an abelian semigroup with identity 0. A set of positives in A is an ordered down-directed set P containing with every r an element r/2 with r/2 + r/2 = r. A continuity space is an abstract set X equipped with a map d : XxX to A satisfying d(x, x) = 0 and d(x, z) d(x, y) + d(y, z). A quasi-uniform space is an abstract set X equipped with a filterbase of binary relations {U} such that each U contains the diagonal as well as for some V{U}. For each rP, the set } is seen to be a quasi-uniform filterbase on X . Indeed, the down-directedness of P ensures that U(r) is a filterbase of oversets of the diagonal and U(r) contains U(r/2)U(r/2). One obtains a uniform filterbase by symmetrization, i.e. by intersecting the U(r) with the U(s) = {(y, x)|d(y, x) <s}.