I'm currently reading Griffith's Introduction to Algebraic Curves. In chapter 2, he proves the following:
Given $C$ an irreducible algebraic curve, and $S$ the set of singularities in the curve, $C^{*}=C\backslash S$ is connected.
I understand that $S$ is a finite set, but I don't see how intuitively how $C$ stays connected when you remove singularities. For example, when taking the curve defined by $x^3-x^2+y^2=0$. This function clearly has a singularity at the origin. When the origin is removed, how is the resulting curve still connected- there is a section to the left of the y axis and a section to the right. Taking the closure of either could not intersect the other. How is the set $C\backslash S$ still connected, intuitively? What part of what I'm saying doesn't make sense?