GδG_\delta-topology and compact cardinals

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

Generically extendible cardinals

In this paper, we study generically extendible cardinal, which is a generic version of extendible cardinal. We prove that the generic extendibility of $\omega_1$ or $\omega_2$ has small consistency strength, but of a cardinal $>\omega_2$ is not. We also consider some results concerning with generic extendible cardinals, such as indestructibility, generic absoluteness of the reals, and Boolean valued second order logic

NOTES ON PRODUCTS OF LINDELÖF SPACES WITH POINTS Gdelta{G_{delta}} (Iterated Forcing Theory and Cardinal Invariants)

In this note, under some extra assumptions, we study some constructions of regular T_{1} Lindelöf spaces with points G_{$delta$} whose product have a large extent