# Example in dimension theory

Could you give me an example of a complete metric space wiht covering dimension $> n$ all of which compact subsets have covering dimension $\le n$?

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In $l^2$ Hilbert space, consider the set $E$ of points with all coordinates rational. Erdös (reference) showed that $E$ has topological dimension $1$. (In separable metric space, all notions of topological dimension coincide.)
Does this $E$ have the property that every compact subset is zero-dimensional? This space (and thus any subset of it) is totally disconnected, and isn't it the case that for compact (metric) spaces, this implies zero-dimensinal?
Thank you, I add "completeness" but Erdös constructs also a complete set ($R_1$). It seems to work, but I need to think a bit. – ε-δ Oct 26 '10 at 17:48
@Willie, If your set $K$ is compact then there is projection $\pi:K\to\mathbb R^n$ to a coordinate n-plane such that preimage of a set of small diameter has small diameter --- from this you get that $\dim K=0$ – ε-δ Oct 26 '10 at 17:58