A closure system is provided for containers used for storing or administering substances in the form of liquids, pastes or powders, which includes a cap, provided with a through-hole, and a closure element, wherein the cap holds the closure element with a force fit and/or form fit on the container, in the area of the container opening that is to be closed, by a catch element present on the container. The end face which surrounds the container opening, and on which the closure element bears, comprises a depression. The closure element is a virus-proof, bacteria-proof and spore-proof film that lies on the end face and, at least in some areas, over the depression. The cap is then fitted in place, an elastomer ring or adhesive ring is arranged between the closure element and the depression and fills the depression.