IV.R215 — The algebraic lemniscate --- no topology assumed

The lemniscate L = S^1 v S^1 is obtained algebraically from Spec of the split-complex algebra R[j]/(j^2 - 1), not by geometric gluing. The topology is the Zariski topology of the spectrum. The smooth circle structure emerges later (Book II, Part V) from the algebraic seed.