Example in dimension theory - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T23:47:52Z http://mathoverflow.net/feeds/question/43680 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/43680/example-in-dimension-theory Example in dimension theory ε-δ 2010-10-26T15:52:20Z 2010-12-30T18:04:42Z <p>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\$?</p> http://mathoverflow.net/questions/43680/example-in-dimension-theory/43688#43688 Answer by Gerald Edgar for Example in dimension theory Gerald Edgar 2010-10-26T16:43:15Z 2010-10-26T16:48:52Z <p><strong>A guess</strong></p> <p>In \$l^2\$ Hilbert space, consider the set \$E\$ of points with all coordinates rational. Erdös (<a href="http://www.jstor.org/pss/1968851" rel="nofollow">reference</a>) showed that \$E\$ has topological dimension \$1\$. (In separable metric space, all notions of topological dimension coincide.) </p> <p>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?</p>