When quoting this document, please refer to the following
URN: urn:nbn:de:0030-drops-1348
URL: http://drops.dagstuhl.de/opus/volltexte/2005/134/
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### Auxiliary relations and sandwich theorems

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### 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.

### BibTeX - Entry

@InProceedings{god_et_al:DSP:2005:134,
author =	{Chris God and Achim Jung and Robin Knight and Ralph Kopperman},
title =	{Auxiliary relations and sandwich theorems},
booktitle =	{Spatial Representation: Discrete vs. Continuous Computational Models},
year =	{2005},
editor =	{Ralph Kopperman and Michael B. Smyth and Dieter Spreen and Julian Webster},
number =	{04351},
series =	{Dagstuhl Seminar Proceedings},
ISSN =	{1862-4405},
publisher =	{Internationales Begegnungs- und Forschungszentrum f{\"u}r Informatik (IBFI), Schloss Dagstuhl, Germany},
address =	{Dagstuhl, Germany},
URL =		{http://drops.dagstuhl.de/opus/volltexte/2005/134},
annote =	{Keywords: Adjoint , auxiliary relation , continuous poset , pairwise completely regular (and pairwise normal) bitopological space , upper (lower) semicontinuous}
}


 Keywords: Adjoint , auxiliary relation , continuous poset , pairwise completely regular (and pairwise normal) bitopological space , upper (lower) semicontinuous Freie Schlagwörter (deutsch): Urysohn relation Seminar: 04351 - Spatial Representation: Discrete vs. Continuous Computational Models Issue Date: 2005 Date of publication: 22.04.2005

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