Families of mutually complementary topologies

Abstract

Several lattices of topologies on an infinite set are considered and bounds are given for the sup of the set of cardinals d such that there is a family of d mutually complementary topologies. Large classes of ℵ 0 {\aleph _0} -topologies are shown to have ℵ 0 {\aleph _0} -complements, and an example is given to prove that complementation is not, in general, a very selective topological operation.