Topological properties of the binary supremum function

Abstract

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

Non-Hausdorff topology

“…[8,9,27]) For a nonempty subset C of a T 0 space X, it is easy to see that C is compact iff ↑C ∈ K X ðÞ . Furthermore, we have the following useful result (see, e.g., [10]). Lemma 2.7.…”