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We show that if CH holds and either (i) there exists an ω 1 -Kurepa tree, or (ii) □ ( ω 2 ) holds, then there are regular T 1 Lindelof spaces X 0 and X 1 with points G δ such that the extent of X 0 × X 1 is strictly greater than 2 ω .

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

In this paper we investigate more characterizations and applications of δ-strongly compact cardinals. We show that, for a cardinal κ, the following are equivalent: (1) κ is δ-strongly compact, (2) For every regular λ ≥ κ there is a δ-complete uniform ultrafilter over λ, and (3) Every product space of δ-Lindelof spaces is κ-Lindelof. We also prove that in the Cohen forcing extension, the least ω 1 -strongly compact cardinal is an exact upper bound on the tightness of the products of two countably tight spaces.

We construct a normal countably tight T 1 space X with t ( X δ ) > 2 ω . This is an answer to the question posed by Dow-Juhasz-Soukup-Szentmiklossy-Weiss [5] . We also show that if the continuum is not so large, then the tightness of G δ -modifications of countably tight spaces can be arbitrary large up to the least ω 1 -strongly compact cardinal.

The mantle is the intersection of all ground models of V. We show that if there exists an extendible cardinal then the mantle is the smallest ground model of V.

Starting from a model with a weakly compact cardinal, we construct a model in which the weak stationary reflection principle for $\omega_2$ holds but the Fodor-type reflection principle for $\omega_2$ fails. So the stationary reflection principle for $\omega_2$ fails in this model. We also construct a model in which the semi-stationary reflection principle holds but the Fodor-type reflection principle for $\omega_2$ fails.

Superstrong cardinals are never Laver indestructible. Similarly, almost huge cardinals, huge cardinals, superhuge cardinals, rank-into-rank cardinals, extendible cardinals, 1-extendible cardinals, 0-extendible cardinals, weakly superstrong cardinals, uplifting cardinals, pseudo-uplifting cardinals, superstrongly unfoldable cardinals, \Sigma_n-reflecting cardinals, \Sigma_n-correct cardinals and \Sigma_n-extendible cardinals (all for n>2) are never Laver indestructible. In fact, all these large cardinal properties are superdestructible: if \kappa\ exhibits any of them, with corresponding target \theta, then in any forcing extension arising from nontrivial strategically, Comment: 19 pages. Commentary concerning this article can be made at http://jdh.hamkins.org/superstrong-never-indestructible. Minor changes in v2