Two Counterexamples Concerning the Scott Topology on a Partial Order

Peter Hertling

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arXiv · 1607.04128

Two Counterexamples Concerning the Scott Topology on a Partial Order

Peter Hertling

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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.

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Peter Hertling. 2016-07-14. Two Counterexamples Concerning the Scott Topology on a Partial Order. https://arxiv.org/abs/1607.04128

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