  Inspired to the work of Ma and Wu\cite{Ma} and Climenhaga\cite{Climenhaga},
we introduce the new nation of topological pressure of a semigroup of maps by
using the Carath\'{e}odory-Pesin structure (C-P structure) with respect to
arbitrary subset in this paper. Moreover, by Bowen's equation, we characterize
the Hausdorff dimension of an arbitrary subset, where the points of the subset
have the positive lower Lyapunov exponents and satisfy a so called tempered
contraction condition.
