Jordan Curve

Definition

Let \(\gamma\) be a continuous path from \([a, b]\) into \(\mathbb{C}\) or \(\mathbb{R}^2\) such that

  1. \(\gamma(a) = \gamma(b)\)
  2. \(\gamma(t_1) \neq \gamma(t_2)\) for all \(t_1, t_2 \in [0, 1)\) satisfying \(t_1 \neq t_2\)

Then \(\gamma\) is called a Jordan curve.

This definition encompasses the fact that the Jordan curve sketch out a joined and closed shape in the complex plane which does not overlap itself. That is, the following is a Jordan curve

while neither of these two are

This yields a distinct inside and outside region, an idea formalised in the Jordan curve theorem.