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Can we talk about a canonical space of dimension $\pi$? Is there anything like $\mathbb R^\pi$?

Have anyone met any fractal of dimension $\pi$?

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There's more than one definition of dimension, which are you interested in? (Some are restricted to the natural numbers) –  Ben Millwood Sep 18 '12 at 13:29
Your ordinary every-day dimension is going to be a cardinal number, but then again it might be one of the numerous dimensions Ben M mentions, which I know nothing about. –  rschwieb Sep 18 '12 at 13:47
I believe he's talking about Hausdorff Dimension, in which case the Hausdorff Dimension Theorem says such spaces exist. –  JSchlather Sep 18 '12 at 15:05
I'm interested in it for every suitable meaning of 'dimension'. –  Berci Sep 18 '12 at 15:54
@JacobSchlather Completely reedited this comment. I thought I remembered what Hausdorff dimension was, but it turns out I forgot a lot :) In any case, $\mathbb{R}^\alpha$ screams vector space to me, and I have no idea how to interpret it as having a strange Hausdorff dimension. –  rschwieb Sep 18 '12 at 16:39
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