  This paper defines and discusses the dimension notion of topological slow
entropy of any subset for Z^d actions. Also, the notion of measure-theoretic
slow entropy for Z^d actions is presented, which is modified from Brin and
Katok [2]. Relations between Bowen topological entropy [3,17] and topological
slow entropy are studied in this paper, and several examples of the topological
slow entropy in a symbolic system are given. Specifically, a variational
principle is proved.
