Hereditarily supercompact spaces

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+¬CH) each separable hereditarily supercompact space is hereditarily separable and hereditarily Lindelöf. This implies that under (MA+¬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]×αD[0,1]×αD of the closed interval and the one-point compactification αD of a discrete space D   of cardinality |D|⩾non(M)|D|⩾non(M) is not hereditarily supercompact (but is Rosenthal compact and uniform Eberlein compact). Moreover, under the assumption cof(M)=ω1cof(M)=ω1 the space [0,1]×αD[0,1]×αD contains a closed subspace X which is first countable and hereditarily paracompact but not supercompact.