Künzi, HansPeter A. ;
Zypen, Dominic van der
The Construction of Finer Compact Topologies
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_1topology is contained in a maximal compact topology and that each sober compact T_1topology which is locally compact or sequential is the infimum of a family of maximal compact topologies.
BibTeX  Entry
@InProceedings{knzi_et_al:DSP:2005:122,
author = {HansPeter A. K{\"u}nzi and Dominic van der Zypen},
title = {The Construction of Finer Compact Topologies},
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 = {18624405},
publisher = {Internationales Begegnungs und Forschungszentrum f{\"u}r Informatik (IBFI), Schloss Dagstuhl, Germany},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2005/122},
annote = {Keywords: Maximal compact , KCspace , sober , USspace , locally compact , sequential , sequentially compact}
}
Keywords: 

Maximal compact , KCspace , sober , USspace , locally compact , sequential , sequentially compact 
Seminar: 

04351  Spatial Representation: Discrete vs. Continuous Computational Models

Issue date: 

2005 
Date of publication: 

22.04.2005 