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The Construction of Finer Compact Topologies Künzi, Hans-Peter A. Zypen, Dominic van der Maximal compact KC-space sober US-space locally compact sequential sequentially compact 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Hans-Peter A. Künzi and Dominic van der Zypen 2005 Is Part Of Dagstuhl Seminar Proceedings, Volume 4351, Spatial Representation: Discrete vs. Continuous Computational Models (2005) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/DagSemProc.04351.18 urn:nbn:de:0030-drops-1224 https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.04351.18 eng https://creativecommons.org/licenses/by/4.0/legalcode