The explanation I have in my notes is
"The affine line $A^1$ is irreducible because it is infinite.
The real line $R$ is not irreducible because it can be written $R = (-\infty, 0] \cup[0, \infty) $"
I understand that in the affine line we "forget" where the origin is, or where we "are" on the line, but whats to stop be picking a random point on the affine line and writing it as the sum of two disjoint sets, just like the real line?
Thanks!
That's $\Bbb R$ in the standard topology that's reducible. The Zarisky topology is far from the standard (in fact, on the affine line it is the cofinite topology).