Journal article: This Note presents a theorem of the existence of the Nash equilibrium for discontinuous games in a topological vector space. We will use an assumption of better reply secure which is stronger then that of Reny. If the payoff function is upper semi-continuous, the two assumptions coincide. Our proof is simple, independent and based on a version of Fan-Browder theorem of existence of maximal element due to Deguire and Lassonde, which is extended to the non-Hausdorf case.