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Mutually complementary families of $ T\sb 1$ 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
DOI: https://doi.org/10.1090/S0002-9939-1995-1301530-9
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 $ {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.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1995-1301530-9
Article copyright: © Copyright 1995 American Mathematical Society

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