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