Mutually complementary families of 𝑇₁ topologies, equivalence relations and partial orders

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 κ \kappa many mutually complementary partial orders (and thus T 0 {T_0} topologies) on κ \kappa and, using this family, build another family of κ \kappa many mutually T 1 {T_1} complementary topologies on κ \kappa . We obtain κ \kappa many mutually complementary equivalence relations on any infinite cardinal κ \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 κ \kappa may not be greater than κ \kappa unless ω > κ > 2 c \omega > \kappa > {2^c} . We show that it is consistent with and independent of the axioms of set theory that there be ℵ 2 {\aleph _2} many mutually T 1 {T_1} -complementary topologies on ω 1 {\omega _1} using the concept of a splitting sequence. We construct small maximal mutually complementary families of equivalence relations.