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Mutually complementary families of ₁ topologies, equivalence relations and partial orders
| Content Provider | Semantic Scholar |
|---|---|
| Author | Steprans, Juris Watson, Stephen |
| Copyright Year | 1995 |
| Abstract | We examine the maximum sizes of mutually complementary families in the lattice of topologies, the lattice of T1 topologies, the semi-lattice of partial orders and the lattice of equivalence relations. We show that there is a family of K many mutually complementary partial orders (and thus To topologies) on K and, using this family, build another family of K many mutually T1 complementary topologies on K. We obtain K many mutually complementary equivalence relations on any infinite cardinal K and thus obtain the simplest proof of a 1971 theorem of Anderson. We show that the maximum size of a mutually T1 complementary family of topologies on a set of cardinality K may not be greater than K unless co < K < 2c . We show that it is consistent with and independent of the axioms of set theory that there be lt2 many mutually T1 -complementary topologies on t1 using the concept of a splitting sequence. We construct small maximal mutually complementary families of equivalence relations. 1. HISTORY AND INTRODUCTION In 1936, Birkhoff published On the combination of topologies in Fundamenta Mathematicae [7]. In this paper, he ordered the family of all topologies on a set by letting T1 < T2 if and only if T1 C T2. He noted that the family of all topologies on a set is a lattice. That is to say, for any two topologies T and a on a set, there is a topology T A a which is the greatest topology contained in both T and a (actually T A a = T n a) and there is a topology T V a which is the least topology which contains both T and a. This lattice has a greatest element, the discrete topology, and a smallest element, the indiscrete topology, whose open sets are just the null set and the whole set. In fact, the lattice of all topologies on a set is a complete lattice; that is to say there is a greatest topology contained in each element of a family of topologies and there is a least topology which contains each element of a family of topologies. A sublattice of this lattice which contains all the Hausdorff spaces is the lattice of T1 topologies. This is also a complete lattice whose smallest element is the Received by the editors June 10, 1992. 1991 Mathematics Subject Classification. Primary 54A10, 06B05; Secondary 03E50, 54A35, 06A12, 08A30. This work has been supported by the Natural Sciences and Engineering Research Council of Canada. @ 1995 American Mathematical Society |
| Starting Page | 2237 |
| Ending Page | 2249 |
| Page Count | 13 |
| File Format | PDF HTM / HTML |
| DOI | 10.1090/S0002-9939-1995-1301530-9 |
| Volume Number | 123 |
| Alternate Webpage(s) | http://www.ams.org/journals/proc/1995-123-07/S0002-9939-1995-1301530-9/S0002-9939-1995-1301530-9.pdf |
| Alternate Webpage(s) | https://doi.org/10.1090/S0002-9939-1995-1301530-9 |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
