$G_{\delta }$-topology and compact cardinals

Abstract

For a topological space $X$, let $X_{\delta }$ be the space $X$ with $G_{\delta }$-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 $G_{\delta }$-topology. We also prove the least measurable cardinal is the supremum of the extents of compact Hausdorff spaces with $G_{\delta }$-topology. For the square of a Lindelöf space, using 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.