A metrizable space $X$ will be called a *generalized Bernstein set* if every closed completely metrizable subspace $C$ of $X$ has cardinality $|C|<|X|$.

It is well-known that the real line contains a (generalized) Bernstein set of cardinality $\mathfrak c$.

Moreover, for every cardinal $\kappa$ with $\kappa^\omega=2^\kappa$ there exist a generalized Bernsten set $X$ of cardinality $|X|=2^\kappa=\kappa^\omega$. In particular, there exists a generalized Bernstein set of cardinality $\beth_{\omega+1}$.

**Question.** What are possible cardinalities of generalized Bernstein sets?
For example, is there a generalized Bernstein set $X$ of cardinality $|X|=\mathfrak c^+$?