Spaces in which compact subsets are closed and the lattice of T1-topologies on a set. (English summary) Comment. Math. Univ. Carolin. 43 (2002), no. 4, 641–652. The set L1(X) of T1-topologies on a set X is a lattice with least element 0, which is the cofinite topology, and greatest element 1, the discrete topology. A T1-complement of τ ∈ L1(X) is a topology σ ∈ L1(X) such that τ ∨ σ = 1 and τ ∧ σ = 0. In a number of papers by several authors it was demonstrated that T1-complements of Hausdorff topologies often are not Hausdorff. The authors study similar questions for spaces which are between Hausdorff and T1-spaces and which are called KC-spaces (or TB-spaces); in these spaces each compact subset is closed. A KC-topology on a countable, infinite set has no T1-complementary KC-topology. Several new properties of KC-spaces are established. For example: a countable KC-space (X, τ) admits a minimal KC-topology τ ′ weaker than τ if and only if there is a weaker sequentia
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