$G_{\delta }$ 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 Lindelöf 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 Lindelöf 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.