G _δ -topology and compact cardinals

For a topological space X, let X _δ be the space X with the G _δ -topology of X. For an uncountable cardinal κ, we prove that the following are equivalent: (1) κ is ω1-strongly compact. (2) For every compact Hausdorff space X, the Lindelöf degree of X _δ is ≤ κ. (3) For every compact Hausdorff space X, the weak Lindelöf degree of X _δ is ≤ κ. This shows that the least ω1-strongly compact cardinal is the supremum of the Lindelöf and the weak Lindelöf degrees of compact Hausdorff spaces with the G _δ -topology. We also prove that the least measurable cardinal is the supremum of the extents of compact Hausdorff spaces with the G _δ -topology. For the square of a Lindelöf space, using a weak G _δ -topology, we prove that the following are consistent: (1) The least ω1-strongly compact cardinal is the supremum of the (weak) Lindelöf degrees of the squares of regular T1 Lindelöf spaces. (2) The least measurable cardinal is the supremum of the extents of the squares of regular T1 Lindelöf spaces.