$G_\delta $-topology and compact cardinals
Volume 246 / 2019
Abstract
For a topological space $X$, let $X_\delta $ be the space $X$ with the $G_\delta $-topology of $X$. For an uncountable cardinal $\kappa $, we prove that the following are equivalent: (1) $\kappa $ is $\omega _1$-strongly compact. (2) For every compact Hausdorff space $X$, the Lindelöf degree of $X_\delta $ is $\le \kappa $. (3) For every compact Hausdorff space $X$, the weak Lindelöf degree of $X_\delta $ is $\le \kappa $. This shows that the least $\omega _1$-strongly compact cardinal is the supremum of the Lindelöf and the weak Lindelöf degrees of compact Hausdorff spaces with the $G_\delta $-topology. We also prove that the least measurable cardinal is the supremum of the extents of compact Hausdorff spaces with the $G_\delta $-topology.
For the square of a Lindelöf space, using a weak $G_\delta $-topology, we prove that the following are consistent: (1) The least $\omega _1$-strongly compact cardinal is the supremum of the (weak) Lindelöf degrees of the squares of regular $T_1$ Lindelöf spaces. (2) The least measurable cardinal is the supremum of the extents of the squares of regular $T_1$ Lindelöf spaces.
