  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 Lindel\"of 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 Lindel\"of 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.
