Classification of metric spaces with infinite asymptotic dimension

We introduce a geometric property called complementary-finite asymptotic dimension (coasdim). Similar to the case of asymptotic dimension, we prove the corresponding coarse invariant theorem, union theorem and Hurewicz-type theorem. Moreover, we show that coasdim(X)≤ω+k implies trasdim(X)≤ω+k and transfinite asymptotic dimension of the shift union sh⋃⨁i=1∞2iZ is no more than ω+1, i.e. trasdim(sh⋃⨁i=1∞2iZ)≤ω+1.