Lang's Complex Analysis contains a proof of the Heine-Borel theorem for $\mathbb{C}^1$, discussed previously on StackExchange.
In the reals, we know that the Heine-Borel theorem holds for $\mathbb{R}^n$. My question is: can Lang's proof be trivially extended to $\mathbb{C}^2$ or $\mathbb{C}^n$ generally?
Best Answer
Well, $\Bbb{C}^n$ is homeomorphic to $\Bbb{R}^{2n}$. So, if you know Heine-Borel for $\Bbb{R}^{2n}$, then you know it for $\Bbb{C}^n$.