$G_\delta$-topology and compact cardinals

TL;DR

This paper establishes deep connections between large cardinal axioms and topological properties of spaces with $G_\delta$-topology, revealing that certain cardinal characteristics are characterized by Lindel"of degrees and extents of these spaces.

Contribution

It proves equivalences between large cardinal properties and topological degrees of $G_\delta$-topologized spaces, linking set theory and topology in novel ways.

Findings

The least $\omega_1$-strongly compact cardinal equals the supremum of Lindel"of degrees of $G_\delta$-topologized compact spaces.

The least measurable cardinal equals the supremum of extents of $G_\delta$-topologized compact spaces.

Consistent existence of spaces where the supremum of Lindel"of degrees of squares matches large cardinal characteristics.

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 Lindel\"of degree of $X_\delta$ is $\le \kappa$. (3) For every compact Hausdorff space $X$, the weak Lindel\"of degree of $X_\delta$ is $\le \kappa$. This shows that the least $\omega_1$-strongly compact cardinal is the supremum of the Lindel\"of and the weak Lindel\"of 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 Lindel\"of 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) Lindel\"of degrees of the squares of regular $T_1$ Lindel\"of spaces. (2) The least measurable cardinal is the supremum of the extents of the squares of regular $T_1$ Lindel\"of spaces.