  A topological space $X$ is called hereditarily supercompact if each closed
subspace of X is supercompact. By a combined result of Bula, Nikiel, Tuncali,
Tymchatyn, and Rudin, each monotonically normal compact Hausdorff space is
hereditarily supercompact. A dyadic compact space is hereditarily supercompact
if and only if it is metrizable. Under (MA + not CH) each separable
hereditarily supercompact space is hereditarily separable and hereditarily
Lindel\"of. This implies that under (MA + not CH) a scattered compact space is
metrizable if and only if it is separable and hereditarily supercompact. The
hereditary supercompactness is not productive: the product [0,1] x \alpha D of
the closed interval and the one-point compactification \alpha D of a discrete
space D of cardinality |D|\ge non(M) is not hereditarily supercompact (but is
Rosenthal compact and uniform Eberlein compact). Moreover, under the assumption
cof(M)=\omega_1 the space [0,1] x \alpha D contains a closed subspace X which
is first countable and hereditarily paracompact but not supercompact.
