  In this paper we introduce a new homology theory devoted to the study of
families such as semi-algebraic or subanalytic families and in general to any
family definable in an o-minimal structure (such as Denjoy-Carleman definable
or $ln-exp$ definable sets). The idea is to study the cycles which are
vanishing when we approach a special fiber. This also enables us to derive
local metric invariants for germs of definable sets. We prove that the homology
groups are finitely generated.
