Let $X$ be a metric space. Recall that the density character, $dc(X)$, is the minimum cardinality of a dense subset of $X$. Assuming that $X$ is a complete metric space what are the possible values of the cardinality, $|X|$?
In the separable case (i.e. $dc(X)=\aleph_0$), the Cantor-Bendixson theorem implies that either $|X| = \aleph_0$ or $|X|=2^{\aleph _0}$, but what about the inseparable case? Clearly we must have $dc(X)\leq |X| \leq dc(X)^{\aleph_0}$, and for any $\kappa=dc(X)$ these bounds are saturated by the discrete metric on $\kappa$ and by the 'standard sequence metric' on $\kappa^\omega$, so whenever $\kappa^{\aleph_0}\leq\kappa^+$ this completely characterizes the possibilities. But depending on set theoretic assumptions and the specific value of $\kappa$ there can be cardinals $\lambda$ strictly between $\kappa$ and $\kappa^{\aleph_0}$, for instance if $2^{\aleph_0}=\aleph_3$, then $\aleph_1<\aleph_2<\aleph_1^{\aleph_0}=2^{\aleph_0}$.
So the question is: For metric spaces $X$ with $dc(X)>\aleph_0$, does ZFC prove any restrictions on the possible values of $|X|$ beyond $dc(X)\leq|X|\leq dc(X)^{\aleph_0}$?
After looking around I was able to find the precise answer to this in Kunen and Vaughan's 'Handbook of Set-Theoretic Topology,' specifically theorem 8.3 (due to Stone) says that for any complete metric space $|X|=dc(X)$ or $|X|=dc(X)^{\aleph_0}$.