Jordan separation from the exponential quotient ($C(J,{\mathbb{C}}^{\ast })/\exp C(J)\cong \mathbb{Z}$)

For a compact set $J$ homeomorphic to a circle, lifting an argument along its parametrization identifies the quotient by continuous exponentials with $\mathbb{Z}$. By planar logarithm factorization it is also free abelian on the bounded complementary components, so exactly one exists. A proper closed arc has trivial exponential quotient, and therefore connected complement. Given a small open arc of $J$ inside any neighborhood of a point, join points from the two complementary regions by a path avoiding the remaining closed arc. The first and last intersections with $J$ occur in the small arc, proving that the neighborhood meets both regions. Thus $J$ is the common boundary of its unique bounded and unbounded regions.