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

A $C_0$ coarse structure for families of pseudometrics and the Higson-Roe functor