Topological properties of the binary supremum function

Abstract We consider the binary supremum function $$\sup :Z\times Z\rightarrow Z$$ sup : Z × → on a sup semilattice Z and its topological properties with respect to Scott topology product topology. It is well known that this continuous $$Z\times . show it open as well. Isbell has constructed several examples of complete lattices such discontinuous Of course, in these cases strictly finer than This raises question whether there exists lattice even construct lattice. Finally, by using any Isbell, we following result: Bounded completeness general not inherited dcpo C ( X , ) functions from where space considered. On other hand, bounded if partial order.