  We introduce the notion of regular finite decomposition complexity of a
metric family. This generalizes Gromov's finite asymptotic dimension and is
motivated by the concept of finite decomposition complexity (FDC) due to
Guentner, Tessera and Yu. Regular finite decomposition complexity implies FDC
and has all the permanence properties that are known for FDC, as well as a new
one called Finite Quotient Permanence. We show that for a collection containing
all metric families with finite asymptotic dimension all other permanence
properties follow from Fibering Permanence.

Metric Geometry;Algebraic Topology;Geometric Topology;K-Theory and Homology