God, Chris ;
Jung, Achim ;
Knight, Robin ;
Kopperman, Ralph
Auxiliary relations and sandwich theorems
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}
}
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Keywords: |
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Adjoint , auxiliary relation , continuous poset , pairwise completely regular (and pairwise normal) bitopological space , upper (lower) semicontinuous |
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Freie Schlagwörter (deutsch): |
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Urysohn relation |
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Seminar: |
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04351 - Spatial Representation: Discrete vs. Continuous Computational Models
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Issue date: |
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2005 |
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Date of publication: |
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22.04.2005 |
