Continuity (Real Parametric Curves)#

Theorem: Jordan's Curve Theorem

Every continuous simple closed parametric curve \(\gamma: [a,b] \to \mathbb{R}^2\) in \(\mathbb{R}^2\) divides \(\mathbb{R}^2\) into two disjoint connected open subsets with the common boundary \(\gamma([a,b])\) such that one is bounded and the other is unbounded.