  We combine aspects of the notions of finite decomposition complexity and
asymptotic property C into a notion that we call finite APC-decomposition
complexity. Any space with finite decomposition complexity has finite
APC-decomposition complexity and any space with asymptotic property C has
finite APC-decomposition complexity. Moreover, finite APC-decomposition
complexity implies property A for metric spaces. We also show that finite
APC-decomposition complexity is preserved by direct products of groups and
spaces, amalgamated products of groups, and group extensions, among other
constructions.
