Abstract
A topology is de Groot dual of another topology, if it has a closed base consisting of all its compact saturated sets. Until 2001 it was an unsolved problem of J. Lawson and M. Mislove whether the sequence of iterated dualizations of a topological space is finite. In this paper we generalize the author's original construction to an arbitrary family instead of a topology. Among other results we prove that for any family $\C\subseteq 2^X$ it holds $\C^{dd}=\C^{dddd}$. We also show similar identities for some other similar and topology-related structures.
BibTeX - Entry
@InProceedings{kovar:DagSemProc.04351.19,
author = {Kovar, Martin},
title = {{The de Groot dual for general collections of sets}},
booktitle = {Spatial Representation: Discrete vs. Continuous Computational Models},
pages = {1--8},
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.dagstuhl.de/opus/volltexte/2005/121},
URN = {urn:nbn:de:0030-drops-1215},
doi = {10.4230/DagSemProc.04351.19},
annote = {Keywords: Saturated set , dual topology , compactness operator}
}
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Keywords: |
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Saturated set , dual topology , compactness operator |
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Collection: |
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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 |
Creative Commons Attribution 4.0 International license (CC BY 4.0)