Mutually complementary families of topologies, equivalence relations and partial orders
Authors:
Juris Steprāns and Stephen Watson
Journal:
Proc. Amer. Math. Soc. 123 (1995), 22372249
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 semilattice 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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 Paul S. Alexandroff, Diskrete Raume, Mat. Sb. 2 (1937), 501519.
 [2]
 , Sur les espaces discrets, C. R. Acad. Sci. Paris 200 (1935), 16491651.
 [3]
 B. A. Anderson, Families of mutually complementary topologies, Proc. Amer. Math. Soc. 29 (1971), 362368. MR 0281141 (43:6860)
 [4]
 B. A. Anderson and D. G. Stewart, complements of topologies, Proc. Amer. Math. Soc. 23 (1969), 7781. MR 0244927 (39:6240)
 [5]
 Bruce A. Anderson, Finite topologies and Hamiltonian paths, J. Combin. Theory Ser. B 14 (1973), 8793. MR 0312447 (47:1004)
 [6]
 , Symmetry groups of some perfect 1factorizations of complete graphs, Discrete Math. 18 (1977), 227234. MR 0463015 (57:2979)
 [7]
 Garrett Birkhoff, On the combination of topologies, Fund. Math. 26 (1936), 156166.
 [8]
 Jason Brown and Stephen Watson, Finite topologies, preorders and their complements; the diameter of the graph (to appear).
 [9]
 , Finite topologies, preorders and their complements; the vertex degree (to appear).
 [10]
 , Mutually complementary partial orders, Discrete Math. 113 (1993), 2739. MR 1212868 (94h:06005)
 [11]
 , Selfcomplementary topologies and preorders, Order 7 (1991), 317328. MR 1120979 (93a:06002)
 [12]
 D. Dikranjan and A. Policriti, Complementation in the lattice of equivalence relations, preprint.
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 Eric K. van Douwen, The integers and topology, Handbook of SetTheoretic Topology (K. Kunen and J. Vaughan, eds.), NorthHolland, Amsterdam, 1984, pp. 111167. MR 776622 (87f:54008)
 [14]
 H. Gaifman, The lattice of all topologies on a denumerable set, Notices Amer. Math. Soc. 8 (1961), 356.
 [15]
 Oystein Ore, Theory of equivalence relations, Duke Math. J. 9 (1942), 573627. MR 0007388 (4:128f)
 [16]
 A. K. Steiner, The lattice of topologies: structure and complementation, Trans. Amer. Math. Soc. 122 (1966), 379398. MR 0190893 (32:8303)
 [17]
 A. W. Tucker, Cell spaces, Ann. of Math. 37 (1936), 92100. MR 1503271
 [18]
 Stephen Watson, Selfcomplementation in countable relations (to appear).
 [19]
 , A completely regular space which is its own complement, Proc. Amer. Math. Soc. (to appear).
 [20]
 , The number of complements, Topology Appl. 55 (1994), 101125. MR 1256214 (94k:54002)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939199513015309
PII:
S 00029939(1995)13015309
Article copyright:
© Copyright 1995
American Mathematical Society
