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Hausdorff Dimension of this set

  1. May 10, 2013 #1
    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})?##
     
  2. jcsd
  3. May 10, 2013 #2
  4. May 13, 2013 #3
    So that gives the ##Hausdorff \ dim## of ##C^c## to be 1. makes sense.
     
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