Two Counterexamples Concerning the Scott Topology on a Partial Order

TL;DR

This paper presents two counterexamples related to the Scott topology on a partial order, demonstrating discontinuity of the supremum function and the non-inheritance of bounded completeness in function spaces.

Contribution

It constructs a complete lattice with a discontinuous supremum function and shows bounded completeness is not preserved in certain function spaces.

Findings

Constructed a complete lattice with a discontinuous supremum function.

Showed bounded completeness is not inherited by the space of continuous functions.

Provided counterexamples to assumptions about Scott topology properties.

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.