What Is the Hausdorff Dimension of \(\mathbb{Y}\)?

[Bachelier]

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 $HD(\mathbb{Y})?$

 

[micromass]

Wikipedia seems to have a nice theorem about this: http://en.wikipedia.org/wiki/Hausdorff_dimension#Self-similar_sets

This theorem implies that the Hausdorff dimension is the solution of $6\left(\frac{1}{3}\right)^s = 1$.

 

[Bachelier]

Wikipedia seems to have a nice theorem about this: http://en.wikipedia.org/wiki/Hausdorff_dimension#Self-similar_sets

This theorem implies that the Hausdorff dimension is the solution of $6\left(\frac{1}{3}\right)^s = 1$.

So that gives the $Hausdorff \ dim$ of $C^c$ to be 1. makes sense.