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



Example in dimension theory
**A guess**


In $l^2$ Hilbert space, consider the set $E$ of points with all coordinates rational. Erdös ([reference](http://www.jstor.org/pss/1968851)) 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?


