Sławomir Turek

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+¬CH) each separable hereditarily supercompact space is hereditarily separable and hereditarily Lindelof. 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.

A decomposition theorem for compact groups with an application to supercompactness

We show that every compact connected group is the limit of a continuous inverse sequence, in the category of compact groups, where each successor bonding map is either an epimorphism with finite kernel or the projection from a product by a simple compact Lie group.As an application, we present a proof of an unpublished result of Charles Mills from 1978: every compact group is supercompact.

Parovičenko spaces with structures

We study an analogue of the Parovičenko property in categories of compact spaces with additional structures. In particular, we present an internal characterization of this property in the class of compact median spaces.

Boolean algebras admitting a countable minimally acting group

The aim of this paper is to show that every infinite Boolean algebra which admits a countable minimally acting group contains a dense projective subalgebra.