Order compatibility for Cauchy spaces and convergence spaces

A Cauchy structure and a preorder on the same set are said to be compatible if both arise from the same quasi-uniform convergence structure onX. Howover, there are two natural ways to derive the former structures from the latter, leading to “strong” and “weak” notions of order compatibility for Cauchy spaces. These in turn lead to characterizations of strong and weak order compatibility for convergence spaces.

Uniformizable Cauchy spaces

A familyCof filters on a setXis uniformizable if there is a uniformity onXsuch thatCis its collection of Cauchy filters. Using the theory of completions and Cauchy continuous maps for Cauchy spaces, we obtain characterizations of uniformizable Cauchy spaces. In particular, given a Cauchy structureConXwe investigate under what conditions the filteru(C)=⋂F∈CF×Fis a uniformity andCis its collection of Cauchy filters. This problem is treated using Cauchy covering systems.