I'm reading the book "Introduction to Differential Geometry" written by J.W. Robbin and D.A. Salamon.
In the page 41 (Lemma 2.4.10) there's the following assertion:
Suppose that $M\subset\mathbb{R}^n$ is a smooth submanifold and $X:M\to \mathbb{R}^n$ is a smooth vector field. Define for all $p\in M$:
$$\color{red}{I_X(p)}:=\bigcup \big\{I\subset \mathbb{R} \, (\text{open interval}):\text{there's a integral curve }\gamma \in M^I\text{ of }X\text{ such that }\gamma (0)=p\big\}.$$
Then for all $p\in M$ there exists an $M$-open neighborhood $U_p\subseteq M$ of $p$ and a constant $\varepsilon _p>0$ such that $(-\varepsilon _p,\varepsilon _p)\times U_p\subseteq D:=\bigcup _{p\in M}I_X(p)\times \{p\}$.
I tried to show that $U_p:=\big\{q\in M:I_X(q)=I_X(p)\big\}$ is open in $M$ because if this is true then that assertion is also true. However I failed and I don't know how to prove that assertion.
Probably this is a consequence of a known theorem about ordinary differential equations, however I didn't study about ODE yet. I say this because, given a parametrization $\psi :V\to \psi [V]\subseteq M$ of $M$, we can define $f:V\to \mathbb{R}^n$ by $f(x):=(d\psi )_x^{-1}\circ X\circ \psi (x)$ (which is smooth) and guarantee the existence of integral curves of $X$ with the system $\begin{cases}\dot{x}=f(x)\\x(0)=p\in \psi [V]\end{cases}$ and using the Picard-Lindelöf Theorem.
Thank you for your attention!