Coarse uniformities on the rationals

Abstract:

There exist uniform spaces $\mu Q$ homeomorphic with the space Q of rational numbers such that every homeomorphic uniform space admits a uniformly continuous homeomorphism upon $\mu Q$. If “homeomorphism” is replaced by “bijection", the resulting weaker property is equivalent to having as completion a Peano continuum. With “homeomorphism", a (countable) dense subspace of a Hilbert cube has the property, but not a dense subspace of an interval.