  The aim of this paper is to provide some new tools to aid the study of
decomposition complexity, a notion introduced by Guentner, Tessera and Yu. In
this paper, three equivalent definitions for decomposition complexity are
established. We prove that metric spaces with finite hyperbolic dimension have
finite (weak) decomposition complexity, and we prove that the collection of
metric families that are coarsely embeddable into Hilbert space is closed under
decomposition. A method for showing that certain metric spaces do not have
finite decomposition complexity is also discussed.
