Hausdorff Dimension of this set

  • Thread starter Bachelier
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  • #1
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Define:

$$\mathbb{Y} = C \times C^{c} \subset \mathbb{R}^{2}$$
where ##C## is the Cantor set and ##C^{c}## is its complement in ##[0,1]##

First I think ##\mathbb{Y}## is neither open nor closed.

Second, the Hausdorff dimension of ##C## is ##\Large \frac{log2}{log3}##. How do we compute the ##HD## of the Cartesian product of sets? For instance ##HD(ℝ^{k})= k## hence can we compute ##HD(\mathbb{Y})?##
 

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