A liquid-dispensing container comprising: a housing adapted to hold a liquid in an interior volume, the housing including an opening defined by an inner edge and an outer edge; and a valve member including an inner support and an outer cover co-molded over the inner support, wherein the inner support is a tubular structure made from a relatively stiff plastic material and has a length that extends almost an entire length of the valve member, the inner support having a ledge engaging the outer edge, wherein the inner support includes four flexible legs separated by recesses, wherein the end of each leg includes a cleat, the cleats being circumferentially spaced from each other and spaced from the ledge, said cleats engaging the inner edge, wherein the inner support includes an exterior surface and an interior surface, wherein the interior surface defines an interior passage extending all the way through the valve member, wherein the outer cover includes an inner section and an outer section integrally formed with the inner section, wherein the inner section is engaged with and covers a relatively small part of the interior surface of the upper end of the inner support, wherein the inner section includes an annular portion and a self-closing valve supported by the annular portion, the self-closing valve including slits that define four flaps that will open when sufficient pressure is applied to the valve and substantially blocking the interior passage through the inner support, wherein the annular portion is engaged with and is secured to the interior surface of the inner support; wherein the outer cover is formed from a material having a lower hardness, lower density, and increased elasticity compared to the material from which the inner support is made. a housing adapted to hold a liquid in an interior volume, the housing including an opening defined by an inner edge and an outer edge; and a valve member including an inner support and an outer cover co-molded over the inner support, wherein the inner support is a tubular structure made from a relatively stiff plastic material and has a length that extends almost an entire length of the valve member, the inner support having a ledge engaging the outer edge, wherein the inner support includes four flexible legs separated by recesses, wherein the end of each leg includes a cleat, the cleats being circumferentially spaced from each other and spaced from the ledge, said cleats engaging the inner edge, wherein the inner support includes an exterior surface and an interior surface, wherein the interior surface defines an interior passage extending all the way through the valve member, wherein the outer cover includes an inner section and an outer section integrally formed with the inner section, wherein the inner section is engaged with and covers a relatively small part of the interior surface of the upper end of the inner support, wherein the inner section includes an annular portion and a self-closing valve supported by the annular portion, the self-closing valve including slits that define four flaps that will open when sufficient pressure is applied to the valve and substantially blocking the interior passage through the inner support, wherein the annular portion is engaged with and is secured to the interior surface of the inner support; wherein the outer cover is formed from a material having a lower hardness, lower density, and increased elasticity compared to the material from which the inner support is made.