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 ] × α D of the closed interval and the one-point compactification αD of a discrete space D of cardinality | D | ⩾ non ( M ) is not hereditarily supercompact (but is Rosenthal compact and uniform Eberlein compact). Moreover, under the assumption cof ( M ) = ω 1 the space [ 0 , 1 ] × α D contains a closed subspace X which is first countable and hereditarily paracompact but not supercompact.