Take the 2-minute tour ×  Could you give me an example of a
complete metric space wiht covering dimension $> n$ all of which
compact subsets have covering dimension $\le n$?  share|improve
this question add comment  1 Answer  up vote 6 down vote
accepted  A guess  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?  share|improve
this answer @Gerald: I can't quite figure out how compactness
can be used to rule out the element on the boundary that Erdos
constructed. Help please? –  Willie Wong Oct 26 '10 at 17:44
Thank you, I add "completeness" but Erdös constructs also a
complete set ($R_1$). It seems to work, b