Hausdorff Dimension of this set

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})?##

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.