TITLE: Nonseparable example in dimension theory?
QUESTION [17 upvotes]: Could you give me an example of a complete metric space with covering dimension $> n$ all of which closed separable subsets have covering dimension $\le n$?

The question closely related to this one.

REPLY [5 votes]: There are completely metrizable spaces $X$ with $\dim(X)=1$ and $\mathrm{ind}(X)=0$, where $\dim$ denotes the covering dimension and $\mathrm{ind}$ the small inductive dimension. The first such example is due to P. Roy (see "nonequality of dimensions for metric spaces", TAMS 1968), but there are many others. My favourite is due to E.K. van Douwen (see "The small inductive dimension can be raised by the adjunction of a single point", Indag.Math. 1973).
Note that if $Y$ is a separable (not necessarily closed) subspace of such an $X$ then $\dim(Y)=0$ since $\dim$ and $\mathrm{ind}$ coincide for separable metrizable spaces. So any such $X$ satisfies what you want for $n=0$.