Jordan arc

Definitions and the statement of the Jordan theorem

A Jordan curve or a simple closed curve in the plane ${\displaystyle \mathbb {R} ^{2}}$ is the image${\displaystyle C}$ of an injectivecontinuous map of a circle into the plane, ${\displaystyle \varphi :S^{1}\to \mathbb {R} ^{2}}$. A Jordan arc in the plane is the image of an injective continuous map of a closed and bounded interval ${\displaystyle [a,b]}$ into the plane. It is a plane curve that is not necessarily smooth nor algebraic.