Welcome to MathOverflow

Question

4

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

edited Oct 28 2010 at 4:57

asked Oct 26 2010 at 15:52

ε-δ
1,028●5●16

Gerald Edgar · Accepted Answer · 2010-10-26 16:48:52Z UTC

6

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?

edited Oct 26 2010 at 16:48

answered Oct 26 2010 at 16:43

Gerald Edgar
15.1k●1●28●67

@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 2010 at 17:44

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 2010 at 17:48

1

@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 2010 at 17:58

1

Yes, complete Erdös-space (all sequences with only irrational coordinates) is complete, separable, dim= 1, and totally disconnected, so this example will do for the complete case. – Henno Brandsma Nov 7 2010 at 14:49

Why does the Erdös space or its irrational variation admit a complete metrics? – Wlodzimierz Holsztynski Mar 1 at 4:29

show 3 more comments

Example in dimension theory

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

1 Answer

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.