Simple closed curve in the plane

Definition

A simple closed curve in the plane or simple loop in the plane is a simple closed curve in the Euclidean plane. In other words, it is a subset of $\mathbb {R} ^{2}$ that satisfies the following equivalent conditions:

• It can be obtained as the image of $S^{1}$ under an injective continuous map
• It is homeomorphic to $S^{1}$

Note that these definitions are equivalent since $S^{1}$ is compact, the plane is Hausdorff, and any injective map from a compact space to a Hausdorff space is an embedding.

Facts

By the Jordan curve theorem, the simple closed curve divides its complement into exactly two components. One of the components is bounded, and is termed the interior of the curve, while the other component is unbounded and is termed the exterior of the curve.