  This paper deepens into the relations between coarse spaces and
compactifications, by defining a $C_0$ coarse structure attached to a family of
pseudometrics. This definition allow us to give a more topological point of
view on the relations between coarse structures and compactifications ---like
the Higson-Roe compactification, corona and functor and the topological coarse
structure attached to a compactification---, define new functors and giving new
relations between them, in particular, some equivalences of categories.
