Complementation in the lattice of regular topologies (Q5905470)

The authors study the question whether a regular topology on a set \(X\) of cardinality \(\omega_ 1\) has a complement in the lattice of all regular topologies on \(X\). They show that there is a complement if suitable partitions, resp. subspaces with complements exist, that every \(T_ 3\)- topology with \(\pi\)-weight \(\omega_ 1\) has a complement and that regular topologies without complements may exist only in rather peculiar spaces.