The Construction of Finer Compact Topologies (extended Abstract of Talk Presented at the Dagstuhl Seminar 04351) Hans-peter A. Künzi (joint Work with Dominic Van Der Zypen)

Definition 1 (compare [2,5,8]) A topological space is called a KC-space provided that each compact set is closed. A topological space is called a U S-space provided that each convergent sequence has a unique limit. Remark 1 Each Hausdorff space (= T 2-space) is a KC-space, each KC-space is a U S-space and each U S-space is a T 1-space (that is, singletons are closed); and no converse implication holds, but each first-countable U S-space is a Hausdorff space. Definition 2 A compact topology on a set X is called maximal compact provided that it is not strictly contained in a compact topology on X. Remark 2 [4] A topological space is maximal compact if and only if it is a KC-space that is also compact. (These spaces will be called compact KC-spaces in the following.) Example 1 A standard example of a maximal compact topology that is not a Hausdorff topology is given by the one-point-compactification of the set of rationals equipped with its usual topology. Indeed we next note that maximal compact spaces can be anti-Hausdorff (= irreducible). A nonempty subspace S of a topological space is called irreducible if each pair of nonempty open sets of S intersects. Furthermore a topological space 1 The article will appear elsewhere under the title " Maximal (sequentially) compact topologies ". The first author acknowledges financial support of the URC of the University of Cape Town, South Africa.