S-Topology On Complete Lattices

During various sorts of topologies on complete lattices, Scott topology and Lawson topology are two very interesting topologies , and have been studied extensively. Open sets of Scott topology and Lawson topology have a common property (S):If U is open and D is a directed , then sup In this paper , we will introduce a new type of topology which we call it S - topology based on the property (S). We will study some basic properties of S - topology and discuss the relation between S - topology and Scott topology and Lawson topology . We proved that S - topology on a continuous lattice L is monoton Hausdorff and zero dimensional normal , also it is locally compact but not compact in general . We also studied the properties of S - continuous maps , some characteriztions of the continuity of lattices were obtained by means of S - topology . In the final , we studied some categorical properties of S - topology and Lawson topology .