Definition:Smooth Path/Simple/Complex Plane

Definition
Let $\gamma: \closedint a b \to \C$ be a smooth path in $\C$.

$\gamma$ is a simple smooth path :
 * $(1): \quad \gamma$ is injective on the half-open interval $\hointr a b$


 * $(2): \quad \forall t \in \openint a b: \map \gamma t \ne \map \gamma b$

That is, if $t_1, t_2 \in \openint a b$ with $t_1 \ne t_2$, then $\map \gamma a \ne \map \gamma {t_1} \ne \map \gamma {t_2} \ne \map \gamma b$.

Also see
Compare with the topological definition of an arc which requires that $\gamma$ is injective on the closed interval $\closedint a b$.

A simple smooth path $\gamma$ is injective on $\closedint a b$ $\gamma$ is not closed.