Commentary on Hans Hahn’s Contributions to the Theory of Curves

Abstract

In 1887, Camille Jordan defined a curve in his Cours d’Analyse, reflecting the usage at that time, as the continuous image of a line segment (see (8), p. 90). He proceeded to formulate what has become known as the Jordan Curve Theorem: A closed curve without multiple points partitions the plane into an inside and an outside (see (8), p. 92). In (32), O. Vehlen had this to say about Jordan’s proof: “It assumes the theorem without proof in the important special case of a simple polygon and of the argument from that point on, one must admit at least that all details are not given.” Based on a system of axioms for geometry, which are equivalent to Hubert’s axioms of order and connection (see (5)), that he pubhshed in an earlier paper (31), Veblen proved in (32) that Jordan’s theorem is true for simple closed polygons. Hahn, in turn, did not find Veblen’s proof binding (“leider erscheint mir aber der Beweis gerade dieses Satzes bei Veblen nicht bindend” — [14], “On the Order Theorems of Geometry” p. 289) and proceeds with his own proof which is also based on Veblen’s system of axioms. The proof is lengthy and tedious but this is unavoidable, considering that the theorem has to be developed from eight axioms with “point” and “between” as the only undefined concepts and with appropriate definitions interjected whenever required. No use whatever is made of continuity. It takes 26 theorems to get finally to theorem 27, which states that every point that does not lie on a simple (closed) polygon can be joined to every point on the polygon by a polygonal path that does not contain any point from the polygon.