The study of dissipative systems has attracted great attention, as dissipation engineering has become an important candidate toward manipulating light in classical and quantum ways. Here, the behavior of a topological system is theoretically investigated with purely dissipative couplings in a synthetic time-frequency space. An imaginary bandstructure is shown, where eigen-modes experience different eigen-dissipation rates during the evolution of the system, resulting in mode competition between edge states and bulk modes. Numerical simulations show that distributions associated with edge states can dominate over bulk modes with stable amplification once the pump and saturation mechanisms are taken into consideration, which therefore points to a laser-like behavior for edge states robust against disorders. This work provides a scheme for manipulating multiple degrees of freedom of light by dissipation engineering, and also proposes a great candidate for topological lasers with dissipative photonics.