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These are homework questions: Give examples of the following spaces

  1. Uncountable metric space with Hausdorff dimension 0.

  2. $\dim X=1$ with Hausdorff dimension 1 measure measure = 0.

I can't think there is any connection between countable and measure. I have a vagure idea that the first example should be somehow modified Cantor set, each time we remove an interval with length = $c_n$, with $\sum c_n=1$. But is this set still uncountable?

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    Cantor-type sets are uncountable no matter what the length of removed intervals. (It's a general theorem about compact sets without isolated points.) Hausdorff dimension is related to covering the set by small sets. In case of Cantor-type constructions, you cover the set by the intervals you have at step N.2012-05-28
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    Thanks Leonid. I still confused on how to calculate the Hausdorff measure of the general cantor set, since it needs to take infimum over all coverings with diameter $\epsilon$ and then let it goes to zero. Why covered by the intervals is enough?2012-05-28
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    The infimum certainly cannot be negative. Thus, if you can find coverings with arbitrarily small sum $\sum (\dots )^d$, it follows that the infimum is zero.2012-05-28

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