# Integral Curves

Let $\alpha, \beta : I \longrightarrow V$ be integral curves in a manifold $M$ then $A = \{ t : \alpha (t) = \beta (t) \}$ is closed.

Proof:

Let $\{t_n\}$ be have a limit $t$ in $I$. Let $\xi : U \longrightarrow \mathbb{R}^m$ be a chart then $\xi (\alpha (t_n))$ converges to $\xi (\alpha (t))$ by continuity. But $\xi (\beta (t_n))$ also converges to $\xi (\alpha (t))$. Hence $\xi (\alpha (t)) = \xi (\beta (t))$ and so $t \in A$. Thus $A$ is closed.