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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.

We prove that for a compact Hausdorff space X, if λ c ( X ) w ( X ) for every infinite cardinal λ w ( X ) and λ c ( X ) cf ( w ( X ) ) for every infinite cardinal λ cf ( w ( X ) ) , then Tikhonov cube [ 0 , 1 ] w ( X ) is a continuous image of X, in particular the cardinality of X is just 2 w ( X ) . As an application of this result, we consider elementary submodel spaces and improve Tall's result in [17] .

In this paper, we study some connections between characters of countably tight spaces of size ω 1 and inaccessible cardinals. A countable tight space is indestructible if every σ-closed forcing notion preserves countable tightness of the space. We show that, assuming the existence of an inaccessible cardinal, the following statements are consistent: (1) Every indestructibly countably tight space of size ω 1 has character ⩽ ω 1 . (2) 2 ω 1 > ω 2 and there is no countably tight space of size ω 1 and character ω 2 . For the converse, we show that, if ω 2 is not inaccessible in the constructible universe L, then there is an indestructibly countably tight space of size ω 1 and character ω 2 .

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

Let κ be a regular uncountable cardinal, and λ be a cardinal greater than κ. Our main result asserts that if (λ< κ)< κ) = λ< κ, then (pκ, λ(NInκ, λ< κ))+ $\longrightarrow$ ((NSκ, λ[λ]< κ)+, NSκ, λs+)3 and (pκ, λ(NAInκ, λ< κ))+ $\longrightarrow$ (NSκ, λs+)3, where NSκ, λs (respectively, NSκ, λ[λ]< κ) denotes the smallest seminormal (respectively, strongly normal) ideal on Pκ (λ), NInκ, λ< κ (respectively, NAInκ, λ< κ) denotes the ideal of non-ineffable (respectively, non-almost ineffable) subsets of Pκ(λ< κ), and pκ, λ: Pκ(λ< κ) → Pκ (λ) is defined by pκ, λ(x) = x ∩ λ.

We study ineffability, the Shelah property, and indescribability of P κ λ when cf(λ)

We introduce a new reflection principle which we call “Fodor-type Reflection Principle” (FRP). This principle follows from but is strictly weaker than Fleissner's Axiom R. For instance, FRP does not impose any restriction on the size of the continuum, while Axiom R implies that the continuum has size ⩽ ℵ 2 . We show that FRP implies that every locally separable countably tight topological space X is meta-Lindelof if all of its subspaces of cardinality ⩽ ℵ 1 are (Theorem 4.3). It follows that, under FRP, every locally (countably) compact space is metrizable if all of its subspaces of cardinality ⩽ ℵ 1 are (Corollary 4.4). This improves a result of Balogh who proved the same assertion under Axiom R. We also give several other results in this vein, some in ZFC, others in some further extension of ZFC. For example, we prove in ZFC that if X is a locally (countably) compact space of singular cardinality in which every subspace of smaller size is metrizable then X itself is also metrizable (Corollary 5.2).