$G_δ$ covers of compact spaces

Santi Spadaro, Paul Szeptycki

Published 2016-05-18Version 1

We prove that in a countably compact weakly Lindelof normal space of countable tightness, every $G_\delta$ cover has a continuum-sized subcollection with a $G_\delta$-dense union and that in a Lindelof space with a base of multiplicity continuum, every $G_\delta$ cover has a continuum sized subcover. We also solve a long standing question due to Arhangel'skii by constructing a compact space which has a $G_\delta$ cover with no continuum-sized ($G_\delta$)-dense subcollection.