License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/DagSemProc.04351.19
URN: urn:nbn:de:0030-drops-1215
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2005/121/
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Kovar, Martin

The de Groot dual for general collections of sets

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04351.KovarMartin5.Paper.121.pdf (0.2 MB)

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}
}

Keywords: Saturated set , dual topology , compactness operator
Collection: 04351 - Spatial Representation: Discrete vs. Continuous Computational Models
Issue Date: 2005
Date of publication: 22.04.2005

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