Maximal (sequentially) compact topologies

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We revisit the known problem whether each compact topology is contained in a maximal compact topology and collect some partial answers to this question. For instance we show that each compact topology is contained in a compact topology in which convergent sequences have unique limits. We also answer a question of D.E. Cameron by showing that each sequentially compact topology is contained in a maximal sequentially compact topology. We finally observe that each sober compact T₁-topology is contained in a maximal compact topology and that each sober compact T₁-topology which is locally compact or sequential is the infimum of a family of maximal compact topologies.