Asymptotic dimension was introduced by Gromov(1993) to study finitely generated
groups, but this definition can be applied for all metric spaces.
There are some generalization of the finite asymptotic dimension for the coarse
geometry, such as asymptotic property C and the finite decomposition property,
which can be generalized to some coarse invariants.
In part 1, we introduce the definition of asymptotic dimension and its
generalizations, and suggest some result of these properties, focused on the
implications.