= Jordan separation from the exponential quotient
{c}
{title2=$C(J,\mathbb C^*)/\exp C(J)\cong\mathbb Z$}
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.
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