Slight confusion in terminology:
What does $S=\{t \in I : \gamma(t)=\hat{\gamma}(t) \}$ closed in $I$ (open interval) by continuity mean? Why is it different from $S$ being a open set? $\gamma, \hat{\gamma}:J \rightarrow M$ is are integral curves on a smooth vector field.
This is the first part of the proof of "Fundamental Theorem on Flows" or the existence of maximal flow on Lee's Smooth Manifolds p. 213.
The reason $S$ is open is because the proof writes:
$\gamma(t_1)=p \implies \gamma(t_1)=\hat{\gamma}(t_1)=p\in M$
Each $p$ has a neighborhood, thus $\gamma, \hat{\gamma}$ agree on open sets $\implies$ $S$ is made of open sets.
If $\gamma$ is continuously differentiable, then $S$ is the inverse image of the closed set $\{0\}$ under the map $t\mapsto \gamma(t)-\gamma'(t)$, hence is closed (as a subset of $I$).