Let $\textbf{$\gamma$}: \mathbb{R} \rightarrow \mathbb{R}^n$ be a smooth curve and let $T \in \mathbb{R}$. We say that $\textbf{$\gamma$}$ is $T$-periodic if $$\textbf{$\gamma$}(t+T)=\textbf{$\gamma$}(t) \text{ for all } t \in \mathbb{R}.$$ If $\textbf{$\gamma$}$ is not constant and is $T$-periodic for some $T\neq 0$, then $\textbf{$\gamma$}$ is said to be closed.
Is the curve $\gamma(t)=\sin(t)$ closed by this definition? How about $\gamma(t)=(t,\sin(t))$?
I believe that the first curve is closed, but the second is not.
For the first one simply take $T=2\pi$.
For the second one notice that it's injective due to its first component, therefore it can never be of that kind unless $T=0$.