$\mathbf {The \ Problem \ is}:$ If $S$ is a countable union of algebraic subsets of $\mathbb C^n$, i.e. a set defined by algebraic equations in the coordinates of $\mathbb C^n$ then show that $P= \mathbb C^n \setminus S$ is poly-line connected i.e. whose any two points can be joined by finitely many broken lines lying in that set .
$\mathbf {My \ approach} :$ Actually,I tried in this way that for any two point $p,q \in P$, we join them by a line segment $L$ and as $S$ is closed, we let two open balls $N_p$ and $N_q$ of $p$ and $q$ such that both of them doesn't meet $S .$
Then for any point $r$ on $L$, not in $S$, we let such a ball around $r$, not meeting $S$ and for points in $S$, we let a $\delta$ ball around them, they cover the connected set $L$, then I think I need to use the finite chain condition on connected sets and poly-line connectedness of $\mathbb C^n \setminus K$ where $K$ is countable .
But, I can't think of the fact where to use the "algebraic set" .
I found a result that such sets are nowhere dense in $\mathbb R^n .$