TITLE: Path connectedness of varieties
QUESTION [17 upvotes]: Let $X$ be a variety.  Then, is $X$ path connected?  And by path connected, I mean any two closed points $P, Q$ on the variety can be connected by the image of a finite number of non-singular curves.

REPLY [7 votes]: 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.