In the affine setting over $\mathbb{C}$, an algebraic set is path-connected in the analytic topology if it is irreducible. Conversely, it is irreducible if and only if it contains a dense open path-connected subset of smooth points.

See the appendix here.

In the affine setting over $\mathbb{C}$, an algebraic set is path-connected in the analytic topology if it is irreducible (in fact, its smooth locus is path-connected too). Conversely, it is irreducible if and only if it contains a dense open path-connected subset of smooth points.

See the appendix here.