Closed manifold

A closed manifold is a topological manifold that is compact and has no boundary. A closed smooth manifold is a smooth manifold whose underlying topological manifold is closed.

Here “closed” does not merely mean closed as a subset of some ambient Euclidean space; it is intrinsic shorthand for compact without boundary. Spheres and tori are closed, while Euclidean space and a closed disk are not—the former is noncompact and the latter has boundary.