Summary: We obtain that the topology on \(Y\) is the largest topology for \(f\) being continuous if and only if \(f\) is a quotient map. Then we have that the topology on \(Y\) is the largest topology for \(f\) being continuous if and only if for any open subset \(U\) of \(Y, \, f^{-1} (U)\) is an open set of \(X\). The results of this paper improve the relation between quotient map and the largest topology.