Two Counterexamples Concerning the Scott Topology on a Partial Order

Abstract

We construct a complete lattice $Z$ such that the binary supremum function $\sup :Z\times Z\to Z$ is discontinuous with respect to the product topology on $Z\times Z$ of the Scott topologies on each copy of $Z$. In addition, we show that 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 $X$ may be any topological space and where on $Z$ the Scott topology is considered.