Let $L$ be a complete lattice. On $L$ we define the so called quasi Scott topology, denoted by $\tau_qSc$. This topology is always larger than or equal to the Scott topology and smaller than or equal to the strong Scott topology. Results concerning the above topology are given. Also, we introduce and investigate the notions of q-continuous and q-algebraic complete lattices. Finally, we give and examine the quasi Lawson topology on a complete lattice.