Metric entropy of homeomorphism on non-compact metric space

Let T: X → X be a uniformly continuous homeomorphism on a non-compact metric space (X,d). Denote by X* = X ∪ {x*} the one point compactification of X and T* : X* → X* the homeomorphism on X* satisfying T*∣X=T and T*x*=x*. We show that their topological entropies satisfy hd(T, X)≥ h(T*, X*) if X is locally compact. We also give a note on Katok's measure theoretic entropy on a compact metric space.