Hereditarily supercompact spaces

TL;DR

This paper investigates hereditarily supercompact spaces, exploring their properties, characterizations, and limitations, including how they behave under various set-theoretic assumptions and product operations.

Contribution

It provides new insights into the structure and properties of hereditarily supercompact spaces, including their characterization in dyadic spaces and behavior under set-theoretic assumptions.

Findings

Hereditary supercompactness is characterized in dyadic compact spaces as equivalent to metrizability.

Product of [0,1] with certain compactifications is not hereditarily supercompact.

Under specific set-theoretic assumptions, there exist hereditarily paracompact spaces that are not supercompact.

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.