Device for closing a container of a liquid to pasty product and refill closed by such a device

A stopper device for a container of a liquid to pasty product, said stopper device comprising:

• a stopper base comprising a stopper wall having a through communication conduit, said stopper wall being intended to close the opening of a container of a liquid to pasty product,
• a stopper movable in the communication conduit between a closed position and an open position,
• wherein the stopper comprises a bottom and a cavity open on one side of the stopper wall and closed by said bottom, the cavity comprising at least one lateral opening, the stopper being arranged with the communication conduit so that:
• in the closed position, said bottom sealingly closes the communication conduit, and
• in the open position, the stopper is partially out of the communication conduit, said at least one lateral opening and said bottom being out of the communication conduit, so that the communication conduit is open in fluid communication through the stopper,
• the stopper comprising a thrust surface on the edges of the said cavity capable of receiving in axial abutment the edges of a first opening of a tube belonging to a dispensing device in the direction of opening of the stopper,
• wherein the stopper wall comprises a cone flaring out from an opening of the communication conduit on a first surface of the stopper wall, the bottom as well as said at least one lateral opening of the stopper being housed in this cone and at a distance from walls of this cone when the stopper is in the open position.