Closed manifold
A compact manifold without boundary.
A closed manifold is a topological manifold that is compact and has no boundary.
Smooth manifolds
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 positive-dimensional Euclidean space and a positive-dimensional closed disk are not—the former is noncompact and the latter has boundary.