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Auxiliary relations and sandwich theorems

Authors Chris God, Achim Jung, Robin Knight, Ralph Kopperman



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Chris God
Achim Jung
Robin Knight
Ralph Kopperman

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Chris God, Achim Jung, Robin Knight, and Ralph Kopperman. Auxiliary relations and sandwich theorems. In Spatial Representation: Discrete vs. Continuous Computational Models. Dagstuhl Seminar Proceedings, Volume 4351, pp. 1-4, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2005)
https://doi.org/10.4230/DagSemProc.04351.8

Abstract

A well-known topological theorem due to Kat\v etov states: Suppose $(X,\tau)$ is a normal topological space, and let $f:X\to[0,1]$ be upper semicontinuous, $g:X\to[0,1]$ be lower semicontinuous, and $f\leq g$. Then there is a continuous $h:X\to[0,1]$ such that $f\leq h\leq g$. We show a version of this theorem for many posets with auxiliary relations. In particular, if $P$ is a Scott domain and $f,g:P\to[0,1]$ are such that $f\leq g$, and $f$ is lower continuous and $g$ Scott continuous, then for some $h$, $f\leq h\leq g$ and $h$ is both Scott and lower continuous. As a result, each Scott continuous function from $P$ to $[0,1]$, is the sup of the functions below it which are both Scott and lower continuous.
Keywords
  • Adjoint
  • auxiliary relation
  • continuous poset
  • pairwise completely regular (and pairwise normal) bitopological space
  • upper (lower) semicontinuous Urysohn relation

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