G_δ covers of compact spaces

We solve a long standing question due to Arhangel’skii by constructing a compact space which has a G_δ cover with no continuum-sized (G_δ)-dense subcollection. We also prove that in a countably compact weakly Lindelöf normal space of countable tightness, every G_δ cover has a c-sized subcollection with a G_δ-dense union and that in a Lindelöf space with a base of multiplicity continuum, every G_δ 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.