Miranda states the following.
Given an irreducible polynomial in $\mathbb{C}[x,y]$, the singular points of its locus of roots $X$ forms a finite set. If we delete these points of $X$ the the resulting open subset is a Riemann surface.
My question is how do we show this is connected. My intuition is that we are removing a finite set of points of something that is like the complex plane, i.e., topologically like a real 2-d manifold.