Are there such a complete metric space $X$ of weight $k<\mathfrak{c}$ ($w(X)=k$) and a family $\{F\_{\alpha}: \alpha<k\}$ of closed subsets of $X$ that $k<|X\setminus \bigcup F\_{\alpha}|<\mathfrak{c}$ holds ?



Are there such a complete metric space X of weight k (w(X)=k) and ....?
As K.P. and Ramiro both point out in the comments, it follows from $\mathsf{CH}$ that the answer is no. I claim that it is also consistent that the answer is yes.


It is consistent that $\mathfrak{c} > \aleph\_2$ and that there is a partition $\mathcal P$ of the Cantor space $2^\omega$ into $\aleph\_2$ closed sets, such that all but $\aleph\_1$ members of $\mathcal P$ are singletons. (See Theorem 3.11 in [this](https://wrbrian.files.wordpress.com/2012/01/ucr.pdf) paper for a proof.)


Let $Y$ be any completely metrizable space of weight $\kappa = \aleph\_1$, and let $X$ be the disjoint sum of $Y$ and the Cantor space. Let $F\_0 = Y$, and let $\{F\_\alpha :\, 1 \leq \alpha < \omega\_1 \}$ be an enumeration of the members of $\mathcal P$ that are not singletons.


