The Topological Complementation Theorem a la Zorn

Paul S. Schnare
1972 Proceedings of the American Mathematical Society  
Steiner's topological complementation theorem is given a short simple proof using Zorn's Lemma. A. K. Steiner [3, Theorem 7.8, p. 397] proved that the lattice of topologies on a fixed set X, denoted 2 or ¿Z{X), is complemented.1 In fact, she showed that each t e 2 has a complement in n=II(Ar), the sublattice of principal topologies. (A topology t e 2 is principal iff each point xeX has a smallest i-neighborhood: Bt{x).) Her proof was quite complicated and although van Rooij [1] gave a simpler
more » ... 1] gave a simpler proof, his proof used both Zorn's Lemma and two applications of transfinite induction. The purpose of this note is to prove Steiner's Theorem via a standard Zornification by the simple trick of suitably adjoining a new point p to X and subsequently discarding it.
doi:10.2307/2038487 fatcat:ezfhbtjdengulgtftqlim353ay