Mutually complementary families of $T_ 1$ topologies, equivalence relations and partial orders
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- by Juris SteprΔns and Stephen Watson PDF
- Proc. Amer. Math. Soc. 123 (1995), 2237-2249 Request permission
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.References
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Additional Information
- © Copyright 1995 American Mathematical Society
- 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