Mutually complementary families of topologies, equivalence relations and partial orders

Authors:
Juris Steprāns and Stephen Watson

Journal:
Proc. Amer. Math. Soc. **123** (1995), 2237-2249

MSC:
Primary 54A10; Secondary 03E50, 04A05, 06C15, 54A35

MathSciNet review:
1301530

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Abstract: We examine the maximum sizes of mutually complementary families in the lattice of topologies, the lattice of topologies, the semi-lattice of partial orders and the lattice of equivalence relations. We show that there is a family of many mutually complementary partial orders (and thus topologies) on and, using this family, build another family of many mutually complementary topologies on . We obtain many mutually complementary equivalence relations on any infinite cardinal and thus obtain the simplest proof of a 1971 theorem of Anderson. We show that the maximum size of a mutually complementary family of topologies on a set of cardinality may not be greater than unless . We show that it is consistent with and independent of the axioms of set theory that there be many mutually -complementary topologies on using the concept of a splitting sequence. We construct small maximal mutually complementary families of equivalence relations.

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DOI:
https://doi.org/10.1090/S0002-9939-1995-1301530-9

Article copyright:
© Copyright 1995
American Mathematical Society