Geodesic
Remarks on Complementation in the Lattice of all Topologies
Canadian journal of mathematics, Tome 18 (1966) no. 1, pp. 83-88

Voir la notice de l'article provenant de la source Cambridge University Press

Our aim is to prove that certain topologies have complements in the lattice of all the topologies on a given set. Lattices of topologies were studied in (1-8). In (7) Hartmanis points out that the lattice of all the topologies on a finite set is complemented and poses the question whether this is so if the set is infinite. A positive answer is given here for denumerable sets. This result was announced in (6). The case of higher powers remains unsettled, although quite a few topologies turn out to have complements. As far as the author knows, no one has proved the existence of a topology that has no complement. 
Gaifman, Haim. Remarks on Complementation in the Lattice of all Topologies. Canadian journal of mathematics, Tome 18 (1966) no. 1, pp. 83-88. doi: 10.4153/CJM-1966-010-4
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