Is the complement of a complex affine algebraic set in an irreducible complex affine algebraic set (path) connected in the euclidean topology?

Question

Let $n \ge 2$, and $V$ be an affine algebraic set in $\mathbb C^n$ and $W$ be an irreducible affine algebraic set in $\mathbb C^n$, with $V \subsetneq W$ ; then is it true that $W \setminus V$ is connected in the Euclidean topology of $\mathbb C^n $ ? Is it path connected in the Euclidean topology ?

I can see that $W \setminus V$ is connected in the Zariski topology of $\mathbb C^n$, but I can't figure out in the Euclidean topology.