  Asymptotic property C for metric spaces was introduced by Dranishnikos as
generalization of finite asymptotic dimension - asdim. It turns out that this
property can be viewed as transfinite extension of asymptotic dimension. The
original definition was given by Radul. We introduce three equivalent
definitions, show that asymptotic property C is closed under products (open
problem stated "Open problems in topology II") and prove some other facts, i.e.
by defining dimension of a family of metric spaces. Some examples of spaces
enjoying countable trasfinite asymptotic dimension are given. We also formulate
open problems and state "omega conjecture", which inspired most part of this
paper.
