The authors show that the self-similar set for a finite family of contractive similitudes (sim-ilarities, i.e., |fi(x) - fi(y)| = ai|x - y|, x,y ∈ RN, where 0 ＜ ai ＜ 1) is uniformly perfectexcept the case that it is a singleton. As a corollary, it is proved that this self-similar set haspositive Hausdorff dimension provided that it is not a singleton. And a lower bound of theupper box dimension of the uniformly perfect sets is given. Meanwhile the uniformly perfectset with Hausdorff measure zero in its Hausdorff dimension is given.