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. <EOS>