Cover (10, 11, 12) configured to be used with a liquid container for realizing a drinking device in conjunction with the liquid container, one side of the cover (10, 11, 12) being an interior side that is intended to have an interior position in the drinking device when the cover (10, 11, 12) is in place on the liquid container, and another side of the cover (10, 11, 12) being an exterior side that is intended to have an exterior position in the drinking device when the cover (10, 11, 12) is in place on the liquid container, wherein the cover (10, 11, 12) is designed to prevent fluid communication therethrough between the interior side and the exterior side in a default condition, and to allow fluid communication therethrough between the interior side and the exterior side in an actuated condition, wherein the cover (10, 11, 12) includes a gasket and a cover body accommodating the gasket, the cover body comprising a liquid passage area and a sealing rim, wherein, in the default condition of the cover (10, 11, 12), the gasket is in sealing contact to the sealing rim of the cover body through a sealing area, wherein the gasket is operable to realize a local interruption of the sealing contact of the gasket to the sealing rim of the cover body, wherein the gasket has at least one structural feature (33, 34) configured to secure the sealing contact of the gasket to the sealing rim of the cover body in the default condition of the cover (10, 11, 12), characterized in that the structural feature of the gasket comprises a closure lip in an area of the gasket that is closer to the interior side of the cover (10, 11, 12) than the sealing area of the gasket, and wherein the cover body comprises a contacting surface area at close range from the closure lip, the contacting surface area of the cover body facing the closure lip of the gasket at a position further from the interior side of the cover (10, 11, 12) than the closure lip, such that the closure lip is adapted to press against the contacting surface area for increasing the extent to which the cover is leakproof. wherein the cover (10, 11, 12) is designed to prevent fluid communication therethrough between the interior side and the exterior side in a default condition, and to allow fluid communication therethrough between the interior side and the exterior side in an actuated condition, wherein the cover (10, 11, 12) includes a gasket and a cover body accommodating the gasket, the cover body comprising a liquid passage area and a sealing rim, wherein, in the default condition of the cover (10, 11, 12), the gasket is in sealing contact to the sealing rim of the cover body through a sealing area, wherein the gasket is operable to realize a local interruption of the sealing contact of the gasket to the sealing rim of the cover body, wherein the gasket has at least one structural feature (33, 34) configured to secure the sealing contact of the gasket to the sealing rim of the cover body in the default condition of the cover (10, 11, 12), characterized in that the structural feature of the gasket comprises a closure lip in an area of the gasket that is closer to the interior side of the cover (10, 11, 12) than the sealing area of the gasket, and wherein the cover body comprises a contacting surface area at close range from the closure lip, the contacting surface area of the cover body facing the closure lip of the gasket at a position further from the interior side of the cover (10, 11, 12) than the closure lip, such that the closure lip is adapted to press against the contacting surface area for increasing the extent to which the cover is leakproof.