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DOI: 10.4230/DagSemProc.04351.8
Auxiliary relations and sandwich theorems

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

Published in: Dagstuhl Seminar Proceedings, Volume 4351, Spatial Representation: Discrete vs. Continuous Computational Models (2005)

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.
Cite as

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)

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@InProceedings{god_et_al:DagSemProc.04351.8,
  author =	{God, Chris and Jung, Achim and Knight, Robin and Kopperman, Ralph},
  title =	{{Auxiliary relations and sandwich theorems}},
  booktitle =	{Spatial Representation: Discrete vs. Continuous Computational Models},
  pages =	{1--4},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2005},
  volume =	{4351},
  editor =	{Ralph Kopperman and Michael B. Smyth and Dieter Spreen and Julian Webster},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/DagSemProc.04351.8},
  URN =		{urn:nbn:de:0030-drops-1348},
  doi =		{10.4230/DagSemProc.04351.8},
  annote =	{Keywords: Adjoint , auxiliary relation , continuous poset , pairwise completely regular (and pairwise normal) bitopological space , upper (lower) semicontinuous Urysohn relation}
}
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