In Example 0.1 of these notes of Gathmann, the author considers the solution set defined by $$C:=\{(x,y)\in \mathbb{C}^2: y^2=(x-1)\cdots(x-2n)\}\subset \mathbb{C}^2$$
Clearly, on the level of sets this solution set is easily described.
Then the author claims that the solutions can be described topologically as forming a sphere with $n-1$ handles with 2 punctures, motivated by a generalization of the construction of Riemann surfaces attached to the function $w=z^2$.
My question is the following: to which topology of $C$ does the above description apply? Is it the subspace topology of the Euclidean topology induced from $\mathbb{C}^2\cong \mathbb R^4$?
Yes, in the regular analytic topology. That is an example of a hyperelliptic curve (at least for $n \geq 2$). These can be used to show that we have curves of arbitrary genus, which you can see from the handles the curve has a riemann surface.