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Which cardinal $\kappa\geq \omega_1$ is critical for the following property:

Let $X\subset \mathbb R$ and $\kappa>|X|\geq \omega_1$. Then there is an uncountable family $\{X_{\alpha}\}$ such that $|X_{\alpha}|=|X|$, $X_{\alpha}$ is a closed subset of $X$ and $X_{\alpha}\cap X_{\beta}=\emptyset$ for $\alpha\neq \beta$.

Clearly $\kappa\geq\mathfrak{q}_0$.

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  • $\begingroup$ If it is true for $\kappa$, then isn't it also true for any smaller $\kappa$, including $\kappa=\omega_1$? Given any set of size $\omega_1$, first extend it to a set of size $\kappa$, get the $X_\alpha$'s, and then cut back down to the original set. Or have I misunderstood? Oh, maybe when you cut down, you get some $X_\alpha$ becoming countable...is that the issue? $\endgroup$
    – Joel David Hamkins
    Commented Jan 26, 2023 at 20:21
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    $\begingroup$ Critical? What do you mean? $\endgroup$
    – Asaf Karagila ♦
    Commented Jan 27, 2023 at 0:18
  • $\begingroup$ perhaps a revised version of the question would be more clear $\endgroup$
    – Alexander Osipov
    Commented Jan 27, 2023 at 4:37
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    $\begingroup$ I think you should refrain from using the word "critical cardinality", and provide a precise definition of kappa. I guess you mean the minimal cardinal such that the property you describe fails for some space of cardinality kappa. Make this precise. $\endgroup$
    – Boaz Tsaban
    Commented Mar 9, 2023 at 10:04

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