  For any $C^1$ diffeomorphism on a smooth compact Riemannian manifold that
admits an ergodic measure with positive entropy, a lower bound of the Hausdorff
dimension for the local stable and unstable sets is given in terms of the
measure-theoretic entropy and the maximal Lyapunov exponent. The mainline of
our approach to this result is under the settings of topological dynamical
systems, which is also applicable to infinite dimensional $C^1$ dynamical
systems.
