Let $(X, T)$ be a topological dynamical system. Denote by $h (T, K)$ and $h^B (T, K)$ the covering entropy and dimensional entropy of $Ksubseteq X$, respectively. $(X, T)$ is called D-{it lowerable} (resp. {it lowerable}) if for each $0le hle h (T, X)$ there is a subset (resp. closed subset) $K_h$ with $h^B (T, K_h)= h$ (resp. $h (T, K_h)= h$); is called D-{it hereditarily lowerable} (resp. {it hereditarily lowerable}) if each Souslin subset (resp. closed subset) is D-lowerable (resp. lowerable). In this paper it is proved that each topological dynamical system is not only lowerable but also D-lowerable, and each asymptotically $h$-expansive system is D-hereditarily lowerable. A minimal system which is lowerable and not hereditarily lowerable is demonstrated.
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