Abstract
We examine the maximum sizes of mutually complementary families in the lattice of topologies, the lattice of T 1 {T_1} topologies, the semi-lattice of partial orders and the lattice of equivalence relations. We show that there is a family of Îş \kappa many mutually complementary partial orders (and thus T 0 {T_0} topologies) on Îş \kappa and, using this family, build another family of Îş \kappa many mutually T 1 {T_1} complementary topologies on Îş \kappa . We obtain Îş \kappa many mutually complementary equivalence relations on any infinite cardinal Îş \kappa and thus obtain the simplest proof of a 1971 theorem of Anderson. We show that the maximum size of a mutually T 1 {T_1} complementary family of topologies on a set of cardinality Îş \kappa may not be greater than Îş \kappa unless Ď > Îş > 2 c \omega > \kappa > {2^c} . We show that it is consistent with and independent of the axioms of set theory that there be âľ 2 {\aleph _2} many mutually T 1 {T_1} -complementary topologies on Ď 1 {\omega _1} using the concept of a splitting sequence. We construct small maximal mutually complementary families of equivalence relations.
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