风机产品的强制密封性

The effect of forcing axioms on the tightness of the $G_δ$ modification

We show that $\mathsf{PFA}$ implies that the tightness $t(X_δ)$ of the $G_δ$-modification of a Fréchet $α_1$-space $X$ is at most $ω_1$, while $\Box(κ)$ implies that there is a Fréchet $α_1$-space with $G_δ$-tightness equal to $κ$. We use the example constructed from $\Box(κ)$ to show that a local version of the bound $t(X_δ)\le 2^{t(X)L(X)}$ does not hold. We also construct, assuming $\mathsf{MA}$, an example of a Fréchet space whose $G_δ$-tightness is larger than $ω_1$.

Remarks on countable tightness

Countable tightness may be destroyed by countably closed forcing. We characterize the indestructibility of countable tightness under countably closed forcing by combinatorial statements similar to the ones Tall used to characterize indestructibility of the Lindelof property under countably closed forcing. We consider the behavior of countable tightness in generic extensions obtained by adding Cohen reals. We show that certain classes of well-studied topological spaces are indestructibly countably tight. Stronger versions of countable tightness, including selective versions of separability, are…

在forcing下某些拓扑性质的保持

Preservation of some topological properties under forcing

本文研究了在forcing下某些拓扑性质的保持问题，证明了在特定游戏中玩家II的获胜策略在集合forcing扩展中保持不变。

SPM Bulletin 12

In this issue we celebrate the appearance of the proceedings of the first SPM Workshop, announce several mathematical breakthroughs, have two extended contributions by Babinkostova, and a new open problem by Kalenda. Contents: Editor's note; Proceedings of the first workshop on Coverings, Selections, and Games in Topology; A five element basis for the uncountable linear orders; Set mapping reflection; The Proper Forcing Axiom, Prikry forcing, and the Singular Cardinals Hypothesis; A solution to the L space problem and related ZFC constructions; Countable Tightness, Elementary Submodels and Hom…

A note on $δ$-strongly compact cardinals

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 $λ\ge κ$ there is a $δ$-complete uniform ultrafilter over $λ$, and (3) Every product space of $δ$-Lindelöf spaces is $κ$-Lindelöf. We also prove that in the Cohen forcing extension, the least $ω_1$-strongly compact cardinal is a precise upper bound on the tightness of the products of two countably tight spaces.