A maximal integral curve is one that cannot be extended to an integral curve on any larger open interval.
Why does it emphasize "open interval"?
I think the definition as follow is better.
A maximal integral curve is one that cannot be extended to an integral curve on any larger interval.
Isn't it?
By the existence theorem for ordinary differential equations a solution always exists on a (possibly small) open neighborhood of any point. Thus it can always be extended to an open interval. If it could be extended to a boundary point of the interval,then it would also extend beyond that to a slightly larger open interval. So, when talking about maximally defined solutions, it makes sense to consider open intervals only.