# Spooky geometry V

The complement of an algebraic set in $A^n(\Bbb{C})$ is path-connected.

1. Let $x,y$ be points in the complement of an algebraic set $X\subseteq A^n(\Bbb{C})$. The (complex!) line joining $x$ and $y$, which is homeomorphic to $\Bbb{C}$, intersects $X$ in at most finitely many points, so there is a path connecting $x$ and $y$ inside it that does not intersect $X$.