# If $X$ is an irreducible variety in $\mathbb{C}^n$, then $\mathbb{C}^n \setminus X$ is path-connected.

My question: Prove that if $$X$$ is an irreducible variety in $$\mathbb{C}^n$$, then $$\mathbb{C}^n \setminus X$$ is path-connected in the standard metric topology.

For every two distinct points $$p,q$$ in $$\mathbb{C}^n \setminus X$$,let $$L$$ denote the line through $$p$$ and $$q$$. I can prove the line is a continuous path but I don't know how to show $$L \subseteq \mathbb{C}^n \setminus X$$ i.e. how to use the condition that $$X$$ is irreducible?

I'm stuck on this question for several hours. Can anyone give me a hint? Thank you !

Hint: Think about the complex line $$\{tp+(1-t)q\in\mathbb{C}^n\mid t\in\mathbb{C}\}\cong\mathbb{C}$$.
• Bigger hint: What is the intersection of $X$ with this line? – user10354138 Jun 10 at 2:06
• I'm confused....$X$ is just an arbitrary irreducible variety and $L$ is the line through $p$ and $q$. We don't have specific "data" about them, how do we even know $X \cap L$? – bbw Jun 10 at 2:12
• $X$ need not be finite, but $L\cap X$ is finite since it is an algebraic set and can't be $L$. Irreducibility doesn't come in here, the result is true for all algebraic sets $X$. – user10354138 Jun 10 at 4:54