Hereditarily supercompact spaces

Abstract

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.Comment: 12 page

Supercompact minus compact is super

“…Since each metrizable space is a collectionwise normal ℵ-space, Theorem 3 implies that metrizable spaces are super. The superness of supercompact spaces is proved in Theorem 2.2 in [4].…”

On critical cardinalities related to $Q$-sets

“…The study of Q-sets was initiated by founders of Set-Theoretic Topology: Hausdorff [20], Sierpiński [39] and Rothberger [38]. Q-Sets are important as they appear naturally in problems related to (hereditary) normal or σ-discrete spaces; see [1], [10], [15], [16], [17], [18], [19], [21], [22], [25], [36], [37], [40], [41].…”

“…Unfortunately, the cardinal q 0 disappeared in the final version of the paper [19]. So, our initial motivation was to collect known information on the cardinal q 0 in order to have a proper reference (in particular, in the paper [1] exploiting this cardinal). Studying this subject we have found a lot of interesting information on the cardinals q 0 and q scattered in the literature.…”