How to force a countably tight, initially ω_1-compact and non-compact space?

A note on the tightness of $G_δ$-modifications

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-Juhász-Soukup-Szentmiklóssy-Weiss. 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.

具有不可分测度的可数紧对偶球

Countably tight dual ball with a nonseparable measure

本文在钻石原理假设下构造了一个紧空间K，使得P(K)可数紧但K上存在不可数Maharam型的Radon测度，从而区分了P(K×K)与P(K)×P(K)的紧性并回答了Pol的问题。

PFA(S)[S] and Locally Compact Normal Spaces

We examine locally compact normal spaces in models of form PFA(S)[S], in particular characterizing paracompact, countably tight ones as those which include no perfect pre-image of omega_1 and in which all separable closed subspaces are Lindelof.

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.

Topological spaces with an $ω^ω$-base

Given a partially ordered set $P$ we study properties of topological spaces $X$ admitting a $P$-base, i.e., an indexed family $(U_α)_{α\in P}$ of subsets of $X\times X$ such that $U_β\subset U_α$ for all $α\leβ$ in $P$ and for every $x\in X$ the family $(U_α[x])_{α\in P}$ of balls $U_α[x]=\{y\in X:(x,y)\in U_α\}$ is a neighborhood base at $x$. A $P$-base $(U_α)_{α\in P}$ for $X$ is called locally uniform if the family of entourages $(U_αU_α^{-1}U_α)_{α\in P}$ remains a $P$-base for $X$. A topological space is first-countable if and only if it has an $ω$-base. By Moore's Metrization Theorem, a …