Hopf-Rinow theorem simple question

Question

Being reading about the Hopf-Rinow theorem in the Do-Carmo book, I got a simple question. The statement is formulated this way:
Let $M$ be a Riemann manifold (connected); assertions are equivalent
a) $\exp_p$ is defined on all $T_p M$
b) $M$ is complete as metric space.

Consider closed (closed as a subset in $\Bbb R^n$ topology) and boarded manifold in $\Bbb R^n$, for example, half of a sphere. Obviously, it is complete as metric space, but $\exp_p$ isn't defined on all $T_p M$. If you need text, I will add screenshots.

Answer 1

My question has already been answered, here is a link Hopf-Rinow Theorem for Riemannian Manifolds with Boundary . Sorry for my caralessness.