I want to be sure with the definitions.
A curve $\gamma(t)=(f_1(t),f_2(t))$ which is $1-1$ is simple (as it does not have $2$ images for the same $t$)
A curve which is simple and closed is a jordan curve? but if it is closed so it is not $1-1$ as $\gamma(a)=\gamma(b)$ for $\gamma:I\to \mathbb{R}^n $where $I=[a,b]$