# Which cardinal $\kappa\geq \omega_1$ is critical for the following property...?

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

• 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? Jan 26 at 20:21
• Critical? What do you mean? Jan 27 at 0:18
• perhaps a revised version of the question would be more clear Jan 27 at 4:37
• 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. Mar 9 at 10:04