Abstract
Let 2 be the lattice of all topologies definable on an arbitrary set £ . Then S is a complete lattice with the trivial topology, {0, JS}, as the least element and the discrete topology, P(E), as the greatest element. The problem of complementation in the lattice 2 has been outstanding for some time although several investigators have provided partial solutions. Hartmanis [6] first showed that S was a complemented lattice if the set E was finite and Gaifman [4] proved 2 was complemented if E was countable. Berri [ l ] , using the results of Gaifman, was able to provide complements for certain special topologies such as a topological group with a dense, nonopen, countable subgroup. I t is the purpose of this paper to introduce the lattice of principal topologies, and to establish tha t the lattice S of all topologies on a set E is complemented. The following theorems are stated without proof. The full details will be published elsewhere.
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