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Recall that the tightness of a topological space $X$ is defined as the least cardinal $\kappa$ such that for every non-closed subset $A$ of $X$ and every point $x \in \overline{A} \setminus A$, there is a set $B \subset A$ such that $|B| \leq \kappa$ and $x \in \overline{B}$.

QUESTION: Is there a ZFC example of a space which is countably tight in some model of ZFC and uncountably tight in some other model?

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  • $\begingroup$ I have added (cardinal-characteristics) - usage for topics like this seems to be in accordance with the tag info. I was also considering (independence-results), but since I wasn't sure, I have just mentioned it here in a comment rather than directly editing the tags. $\endgroup$
    – Martin Sleziak
    Commented Feb 23, 2022 at 13:45
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    $\begingroup$ Let $X,Y$ be (connected) CW complexes. Then $X\times Y$ is a CW complex iff it is a k-space iff it is sequential iff it has countable tightness (this is due to Tanaka). On the other hand, $X\times Y$ is a CW complex iff one of $X,Y$ has countably many cells and the other has fewer than $\mathfrak{b}$ many cells (this is due to Brooke-Taylor). Here $\mathfrak{b}$ is the bounding number. Thus $\aleph_1\leq\mathfrak{b}\leq2^{\aleph_0}$, and there are models of ZFC in which either or both inequalities are strict, or both are equality. $\endgroup$
    – Tyrone
    Commented Feb 23, 2022 at 23:14
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    $\begingroup$ In the simplest case $(\bigvee_\omega I)\times\bigvee_{\omega_1}I)$ has countable tightness iff $\mathfrak{b}>\aleph_1$ (here $I=[0,1]$ is the unit interval with wedgepoint $0$). $\endgroup$
    – Tyrone
    Commented Feb 23, 2022 at 23:15
  • $\begingroup$ Thank you! Actually, the special case you mentioned in your second comment already appears in Gary Gruenhage, "K-spaces and Products of Closed Images of Metric Space", Proceedings of the American Mathematical Society, Vol. 80, n.3 (1980), 478--482. $\endgroup$
    – Santi Spadaro
    Commented Feb 25, 2022 at 9:46
  • $\begingroup$ Tyrone, can you add your comment as an answer? $\endgroup$
    – Santi Spadaro
    Commented Mar 4, 2022 at 9:11

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