THE NUMBER OF COMPLEMENTS IN THE LATTICE OF TOPOLOGIES ON A FIXED SET

In 1936, Birkhoff ordered the family of all topologies on a set by inclusion and obtained a lattice with 1 and 0. The study of this lattice ought to be a basic pursuit both in combinatorial set theory and in general topology. In this paper, we study the nature of complementation in this lattice. We say that topologies tau and sigma are complementary if and only if tau AND sigma = 0 and tau OR sigma = 1. For simplicity, we call any topology other than the discrete and the indiscrete a proper topology. Hartmanis showed in 1958 that any proper topology on a finite set of size at least 3 has at least two complements. Gaifman showed in 1961 that any proper topology on a countable set has at least two complements. In 1965, Steiner showed that any topology has a complement. The question of the number of distinct complements a topology on a set must possess was first raised by Berri in 1964 who asked if every proper topology on an infinite set must have at least two complements. In 1969, Schnare showed that any proper topology on a set of infinite cardinality kappa has at least kappa distinct complements and at most 2(2kappa) many distinct complements. By exhibiting examples of topologies on a set of cardinality kappa which possess exactly kappa complements, exactly 2kappa complements and exactly 2(2kappa) complements, Schnare showed under the generalized continuum hypothesis that there are exactly three values for the number of complements of a topology on an infinite set. His paper is the origin of the present paper. This paper has three main purposes. First, to completely answer the problem of establishing the exact number of complements of a topology on a set of cardinality N(n) by showing that at most 2n + 4 values can be obtained. Second, to show that all topologies on a set of cardinality kappa, except for some simple and easy to describe ones, have at least 2kappa many complements. This improves the lower bound given by Schnare in 1969. Third, to ask a specific question about the number of complements in the lattice of topologies on a set of singular cardinality which roughly captures the open remnant of Berri's question. This paper is completely self-contained.