The following theorem is called the Jordan Curve Theorem.

Definition

Let \( c \) be a loop⭕️ in the plane (also known as a Jordan curve or simple closed curve). The complement of \( c \) in \( \mathbb{R}^2 \), denoted \( \mathbb{R}^2-c \), consists of a bounded domain (the interior) and an unbounded domain (the exterior), with the boundary of both domains being \( c \). If we take one point from the interior and one from the exterior, any arc connecting them must intersect \( c \).

Although this theorem may seem intuitively obvious, proving it for general loops is difficult and requires knowledge of topology.

Example

Now, we have stated this for \( \mathbb{R}^2 \), but does it hold in \( \mathbb{R}^3 \)? The answer is no. It is clear that for a loop in \( \mathbb{R}^3 \), there is no well-defined interior or exterior.

When we consider loops on surfaces, the theorem holds on the sphere \( S^2 \). However, as mentioned in the article on simply connected, the distinction between interior and exterior has no meaning here.

There are loops on the torus🍩 where the theorem does not hold. For example, a loop that runs along the latitude of the torus (circling around the outside of the donut) does not divide the torus into an interior and exterior. This loop does not split the torus into distinct domains, meaning that the Jordan Curve Theorem does not apply.