On Bowen's definition of topological entropy

About 5 years ago, Dai, Zhou and Geng proved the following result. If $X$ is a metric compact space and $f:X\to X$ a Lipschitz continuous map, then the Hausdorff dimension of $X$ is bounded from below by the topological entropy of $f$ divided by the logarithm of its Lipschitz constant. We show that this is a simple consequence of a 30 years old Bowen's definition of topological entropy for noncompact sets. Moreover, a modification of this definition provides a new insight into the entropy of subshifts of finite type.