# Hausdorff Dimension of this set

1. May 10, 2013

### 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 $HD(\mathbb{Y})?$

2. May 10, 2013

### micromass

3. May 13, 2013

### Bachelier

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