Mutually complementary families of $T_ 1$ topologies, equivalence relations and partial orders

Abstract:

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