Jordan Curve Theorem states: Any simple closed curve c on the 2-sphere separates it into two components U and V. Each component has boundary c. Thinking of the 2-sphere as the compactification of the plane, the theorem seems intuitively plausible. However, the proof is nontrivial because of theorem's generality.
From the right picture, it is clear that the presented curve is both simple and closed. However, the inside and outside are not readily obvious in the left picture. #geometric_topology #Jordan_curve_theorem #simple_closed_curve #topology