TITLE: The graph of xy = 1 is connected or not
QUESTION [30 upvotes]: The graph of $xy = 1$ in $\Bbb C^2$ is connected. True or false?
I know that it is not connected in $\Bbb R^2$, but what is the case of $\Bbb C^2$?

REPLY [10 votes]: Suppose that $w_1,w_2,z_1,z_2\in\Bbb C$ are such that $w_iz_i=1$ for $i=1,2$. There are positive $r_i,s_i\in\Bbb R$ and $\theta_i,\varphi_i\in[0,2\pi)$ such that $w_i=r_ie^{i\theta_i}$ and $z_i=s_ie^{i\varphi_i}$.
Show that the set in question contains a path from $\langle w_1,z_1\rangle$ to $\langle w_2,z_2\rangle$ consisting of a linear path from $\langle w_1,z_1\rangle$ to $\langle r_2e^{i\theta_1},s_2e^{i\varphi_1}\rangle$ followed by a path from $\langle r_2e^{i\theta_1},s_2e^{i\varphi_1}\rangle$ to $\langle w_2,z_2\rangle$ that takes each coordinate around a circular arc in its copy of $\Bbb C$.
Once you’ve done this, you’ll have shown that the set is path connected and a fortiori connected.