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I'm reading a (dynamical systems) paper in which topological dimension figures. In my situation, I'm trying to compute the topological dimension of the subset of the $d$-dimensional torus consisting of points with rational coordinates. I've looked at the online resources that I could find, but it seemed that they were geared towards computing the dimension of compact sets (being expressed in terms of multiplicities of finite open covers etc). Other web sites asserted that all countable sets had dimension 0, but they didn't seem to justify this in any way.

If anyone can point me in the right direction I'd be very grateful.

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The dimension of $\Bbb Q^n$ is $0$ with respect to any of the three main topological notions of dimension. The same is true of any non-empty subset. – Brian M. Scott May 8 '12 at 5:21
Thanks. Is this a simple consequence of the definition, or is it a deeper fact? (it was far from obvious to me from the definitions I saw) – anthonyquas May 8 '12 at 6:45
It's very straightforward for small inductive dimension. It takes a bit more work for large inductive dimension and covering dimension. – Brian M. Scott May 8 '12 at 6:51

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