I proved that $\mathcal H^1 (\mathbb R^2)=\mathbb 0$ and if $\{ P_1,...,P_n\ |P_i\neq P_j,\ i\neq j \}$ is an $\mathbb R^2$ subset, $\mathcal H^1 (\mathbb R^2\setminus\{P_1,...,P_n\})=\mathbb R^n $. Thus I proved this fact: $\mathcal H^1 (\mathbb R^2\setminus(\mathbb Z×\{0\})) =\mathbb R ^ \mathbb N =\mathbb R^{\oplus \mathbb N}$
Now let's denote with $\mathcal T $ Cantor set; we know $\mathcal T $ is closed, thus its complement is open. I'm eventually wondering what $\mathcal H^1 (\mathbb R^2\setminus \mathcal (T×\{0\}) )$ is.