In Do Carmo' Differential Geometry book, the author defines regular curves as follows:
A regular curve in $\mathbb{R}^3$ is a subset $C \subseteq \mathbb{R}^3$ with the following property: For each point $p \in C$ there is a neighborhood $V$ of $p$ in $\mathbb{R}^3$ and an infinite differentiable homeomorphism $\alpha: I \subseteq \mathbb{R} \rightarrow V \cap C$ ($I$ is an open interval) such that $\alpha'(t)\neq 0$ for all $t \in I$.
He also defines parametrized regular curves as follows
A parametrized regular curve is a function $\gamma: I \rightarrow \mathbb{R}^3$, such that $I$ is an open interval, $\gamma$ is infinitely differentiable and $\gamma'(t)\neq 0$.
Here is my question:
Let $C \subset \mathbb{R}^3 $ a regular connected curve. Does there exist a parametrized regular curve $\gamma: I \rightarrow \mathbb{R}^3$ such that $\gamma(I)=C$?
Yes, for example a circle minus one point $X=C(O,1)\setminus(0,-1)$ has parametrization $t\mapsto(\cos(t),\sin(t))$ which is a homeomorphism from $]-\pi,\pi[$ to $X$.