What is an example of a connected Riemannian manifold containing a non-compact closed bounded set?
By the Hopf-Rinow Theorem, I know that the closed bounded sets of a connected Riemannian manifold are compact if and only if the manifold is complete. So I have to pick a non complete Riemannian manifold. Let's say I choose the punctured sphere or an open hemisphere with the induced metric from Euclidean space. "Where is" that closed bounded subset of the sphere which is not compact? Or am I in the wrong track?