Coarse metric and uniform metric

Abstract

We introduce the notion of coarse metric. Every coarse metric induces a coarse structure on the underlying set. Conversely, we observe that all coarse spaces come from a particular type of coarse metric in a unique way. In the case when the coarse structure E on a set X is defined by a coarse metric that takes values in a meet-complete totally ordered set, we define the associated Hausdorff coarse metric on the set P0(X) of non-empty subsets of X and show that it induces the Hausdorff coarse structure on P0(X).On the other hand, we define the notion of pseudo uniform metric. Each pseudo uniform metric induces a uniform structure on the underlying space. In the reverse direction, we show that a uniform structure U on a set X is induced by a map d from X×X to a partially ordered set (with no requirement on d) if and only if U admits a base B such that B∪{⋂U} is closed under arbitrary intersections. In this case, U is actually defined by a pseudo uniform metric. We also show that a uniform structures U comes from a pseudo uniform metric that takes values in a totally ordered set if and only if U admits a totally ordered base.Finally, a valuation ring will produce an example of a coarse and pseudo uniform metric that take values in a totally ordered set.