A surface is closed if it has no boundary curves - a boundary curve being defined as a curve which separates the inside of a surface from the outside. If a surface has one or more boundary curves it is said to be open.

A sphere, a cube (which is homeomorphic to a sphere), and a torus with n holes are all closed surfaces.

A cube is closed because it has edges but no boundary.

A hollow cylinder or a cylinder open at one end (but having no volume) is an open surface but a solid cylinder is a closed surface.

A mobius band is an open surface with one boundary curve.

A klein bottle is open and one sided.