These are homework questions: Give examples of the following spaces
Uncountable metric space with Hausdorff dimension 0.
$\dim X=1$ 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?