I've wondered if the following is an example of 1, but I'm not expert enough in algebra to know.
The set of smooth points of an irreducible complex projective variety is connected in the classical topology.
The argument I know goes like this: Suppose it were disconnected, a disjoint union of $A$ and $B$, say. These are locally analytic sets. Then, by a theorem of Remmert and Stein, their closures $\overline{A}$ and $\overline{B}$ are analytic sets. Then, by Chow's part of the GAGA principle, $\overline{A}$ and $\overline{B}$ are varieties, and the original variety is not irreducible.
I've always wondered if you can avoid the Remmert-Stein theorem in the middle (without using Hironaka's theorem).
