Definition: Jordan Arc (Simple Curve)

A curve $f:I\to\mathbb R^n.$ is called simple or a Jordan arc, if it is injective, i.e. if for any two $x,y\in I$ from $f(x)=f(y)$ it follows that $x=y$ .

Example

Please note that the injectivity of $f$ assures that there are no two points $x,y$ with $x\neq y$ , for which $f(x)=f(y)$ . In other words, the curve has no "intersection points". For a curve in the plane $\mathbb R^2$ this can be visualized as follows:

simple_curve