TheoremProved

Jordan curve theorem

Statement

Let $\gamma$ be a simple closed curve in the plane $\mathbb{R}^2$ (a continuous injective image of a circle). Then $\mathbb{R}^2 \setminus \gamma$ has exactly two connected components, a bounded interior and an unbounded exterior, and $\gamma$ is the boundary of each.

Why is it true?

It looks obvious for a circle or a wiggly loop drawn by hand: an insect inside can't reach the outside without crossing the curve. The theorem is hard precisely because 'simple closed curve' also allows wild, nowhere-smooth, infinitely jagged loops (such as the boundary of the Koch snowflake), for which 'inside' and 'outside' still need a rigorous meaning and proof.

Proof sketch

Modern rigorous proofs (first given by Veblen in 1905, after Jordan's own 1887 argument was found to have gaps) proceed by approximating the curve by polygons and using purely combinatorial parity arguments (a ray from a point crosses the curve an even or odd number of times depending on whether the point is outside or inside), or, in the algebraic-topology approach, by computing that $\mathbb{R}^2$ minus the curve has exactly two path components via Alexander duality / reduced homology of the complement.

Topics that use this theorem

Topological spaces, compactness and connectedness

Related theorems

Borsuk–Ulam theorem Classification of closed surfaces

Step-by-step proofs

No step-by-step proof yet for this theorem.

References

Oswald Veblen (1905). Theory of Plane Curves in Non-Metrical Analysis Situs
James R. Munkres (2000). Topology