$${G_\delta}$$ G δ covers of compact spaces

Abstract

We 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. We also prove that in a countably compact weakly Lindelof normal space of countable tightness, every \({G_\delta}\) cover has a \({\mathfrak{c}}\)-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 finally apply our results to obtain a bound on the cardinality of homogeneous spaces which refines De la Vega’s celebrated theorem on the cardinality of homogeneous compacta of countable tightness.