Consider the surface $S$ in $\Bbb R^3$ defined by the equation $z^{2} -xy = 0$. What is the number of connected components of its complement (in $\Bbb R^3$)?
First of all I have to Find the points where the equation $z^{2} - xy = 0$ is not satisfied. These points lie on either the $xy$-plane $(z=0)$ or the double cone with vertex at the origin and axis along the $z$-axis $(z = \pm \sqrt{xy})$.
And then I have to Determine whether or not each of these sets ($xy$-plane and double cone) is connected.
If both sets are connected, then the complement of the surface $S$ in $\Bbb R^3$ has two connected components: one inside the double cone and one outside. If one or both sets are not connected, then the complement has more than two connected components. Is it in right way .. please guide me from here