An invariant of metric spaces under bornologous equivalences

Abstract

We define bornologous equivalence between metric spaces, which is a morerestrictive equivalence than coarse equivalence. It requires spaces to have thesame cardinality. We define an invariant of metric spaces under bornologousequivalences. The invariant is essentially the number of ways that sequencescan go to infinity in a space. This invariant is only for a certain class of spacesthat we call sigma stable.