  Hurewicz's dimension-raising theorem states that for every n-to-1 map f : X
\to Y, dim Y =< dim X + n holds. In this paper we introduce a new notion of
finite-to-one like map in a large scale setting. Using this notion we formulate
a dimension-raising type theorem for the asymptotic dimension and the
asymptotic Assouad-Nagata dimension. It is also well-known as Hurewicz's
finite-to-one mapping theorem that dim X =< n if and only if there exists an (n
+ 1)-to-1 map from a 0-dimensional space onto X. We formulate a finite-to-one
mapping type theorem for the asymptotic dimension and the asymptotic
Assouad-Nagata dimension.
