Convergence On Quasi-continuous Domain

In this paper the S*-convergence and quasi-liminf convergence of a net on quasi-continuous domain is introduced. The concept of S*-coiivergence and quasi-liminf convergence generalize the S-convergence and and the liminf convergence on domain, respectively. The main result is that for a dcpo L the S*-convergence is topological if and only if L is quasi-continuous and the topology generated by it coincides with the Scott topology. And under an assumption, the quasi-liminf convergence is topological if and only if L is quasi-continuous and the topology generated by it coincides with the Lawson topology.