The Jordan Curve Theorem

Definition: Let $f : S^1 \to \mathbb{R}^2$ be an embedding. A Jordan Curve or Plane Simple Closed Curve is the image $f(S^1)$.

In other words, if $f : S^1 \to \mathbb{R}^2$ is an embedding of the circle in $\mathbb{R}^2$, then $f(S^1)$ will be a closed loop that does not intersect itself nontrivially. Some examples of Jordan curves are illustrated below:

Before we state the Jordan curve theorem, we will first state an important lemma that can be used to prove the result.

Lemma 1: Let $\alpha, \beta : [0, 1] \to [0, 1] \times [0, 1]$ be paths and suppose that: a) $\alpha(0) \in \{ 0 \} \times [0, 1]$. b) $\alpha(1) \in \{ 1 \} \times [0, 1]$. c) $\beta(0) \in [0, 1] \times \{ 0 \}$. d) $\beta(1) \in [0, 1] \times \{ 1 \}$. Then $\alpha(I) \cap \beta (I) \neq \emptyset$, that is, the graphs of $\alpha$ and $\beta$ intersect in at least one point on $[0, 1] \times [0, 1]$.

We now state the Jordan curve theorem.

Theorem (The Jordan Curve Theorem): Let $f : S^1 \to \mathbb{R}^2$ be an embedding so that $f(S^1)$ is a Jordan curve. Then $\mathbb{R}^2 \setminus f(S^1)$ has two components - one of which is bounded and the other of which is unbounded.

Intuitively, the Jordan curve theorem is clear but it actually takes quite a bit of work to prove. We will omit the proof for the time being.