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.04351.18
URN: urn:nbn:de:0030-drops-1224
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2005/122/
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Künzi, Hans-Peter A. ; Zypen, Dominic van der

The Construction of Finer Compact Topologies

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04351.KuenziHansPeterAlbert1.ExtAbstract.122.pdf (0.1 MB)

Abstract

It is well known that each locally compact strongly sober topology is contained in a compact Hausdorff topology; just take the supremum of its topology with its dual topology. On the other hand, examples of compact topologies are known that do not have a finer compact Hausdorff topology.
This led to the question (first explicitly formulated by D.E. Cameron) whether each compact topology is contained in a compact topology with respect to which all compact sets are closed. (For the obvious reason these spaces are called maximal compact in the literature.)
While this major problem remains open, we present several partial solutions to the question in our talk. For instance we show that each compact topology is contained in a compact topology with respect to which convergent sequences have unique limits. In fact each compact topology is contained in a compact topology with respect to which countable compact sets are closed. Furthermore we note that each compact sober T_1-topology is contained in a maximal compact topology and that each sober compact T_1-topology which is locally compact or sequential is the infimum of a family of maximal compact topologies.

BibTeX - Entry

@InProceedings{kunzi_et_al:DagSemProc.04351.18,
  author =	{K\"{u}nzi, Hans-Peter A. and Zypen, Dominic van der},
  title =	{{The Construction of Finer Compact Topologies}},
  booktitle =	{Spatial Representation: Discrete vs. Continuous Computational Models},
  pages =	{1--5},
  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/2005/122},
  URN =		{urn:nbn:de:0030-drops-1224},
  doi =		{10.4230/DagSemProc.04351.18},
  annote =	{Keywords: Maximal compact , KC-space , sober , US-space , locally compact , sequential , sequentially compact}
}

Keywords: Maximal compact , KC-space , sober , US-space , locally compact , sequential , sequentially compact
Collection: 04351 - Spatial Representation: Discrete vs. Continuous Computational Models
Issue Date: 2005
Date of publication: 22.04.2005

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