MUTUALLY COMPLEMENTARY FAMILIES OF T-1 TOPOLOGIES, EQUIVALENCE-RELATIONS AND PARTIAL ORDERS

We examine the maximum sizes of mutually complementary families in the lattice of topologies, the lattice of T-1 topologies, the semi-lattic e of partial orders and the lattice of equivalence relations. We shaw that there is a family of kappa many mutually complementary partial or ders (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 relatio ns on any infinite cardinal kappa and thus obtain the simplest proof o f a 1971 theorem of Anderson. We show that the maximum size of a mutua lly T-1 complementary family of topologies on a set of cardinality kap pa 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 N-2 many mutually T-1-complementary topologies on or us ing the concept of a splitting sequence. We construct small maximal mu tually complementary families of equivalence relations.