  Let $(X, T)$ be a topological dynamical system (TDS), and $h (T, K)$ the
topological entropy of a subset $K$ of $X$. $(X, T)$ is {\it lowerable} if for
each $0\le h\le h (T, X)$ there is a non-empty compact subset with entropy $h$;
is {\it hereditarily lowerable} if each non-empty compact subset is lowerable;
is {\it hereditarily uniformly lowerable} if for each non-empty compact subset
$K$ and each $0\le h\le h (T, K)$ there is a non-empty compact subset
$K_h\subseteq K$ with $h (T, K_h)= h$ and $K_h$ has at most one limit point.
  It is shown that each TDS with finite entropy is lowerable, and that a TDS
$(X, T)$ is hereditarily uniformly lowerable if and only if it is
asymptotically $h$-expansive.
