I'm interested in compactifications of some non-separable metric spaces of functions. For concreteness, consider $D[0,1]$, the space of right-continuous real-valued functions on $[0,1]$ with left limits, and give it the uniform (supremum) norm $\|f\|_\infty=\sup_t |f(t)|$.
The Alexandroff one-point compactification $D^*$ of $D[0,1]$ is compact, and $(D[0,1],\,\|\cdot\|_\infty)$ is a dense subspace of it. Is $D^*$ separable?
If it isn't, $D^*$ is a somewhat interesting example of a compact space that isn't separable (this question)
If it is, $(D[0,1],\,\|\cdot\|_\infty)$ and $D^*$ are an interesting example of a non-separable subspace of a separable space (this question)
But I have no idea which.
