Definition:Smooth Path/Simple/Complex Plane

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Definition

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

$\gamma$ is a simple smooth path if and only if:

$(1): \quad \gamma$ 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 $\closedint a b$ if and only if $\gamma$ is not closed.


Sources