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> Could you give me an example of a complete metric space with covering dimension $> n$ all of which closed separable subsets have covering dimension $\le n$?
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The question closely related to [this one](https://mathoverflow.net/questions/43680).



Nonseparable example in dimension theory?
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 $n=0$.


