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The Topological Complementation Theorem a la Zorn

1972
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Proceedings of the American Mathematical Society
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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

doi:10.2307/2038487
fatcat:ezfhbtjdengulgtftqlim353ay