Resumen de Transversal and T1-Independent Topologies

Richard G. Wilson , Mikhail G. Tkachenko , Dmitri Shakhmatov

• A pair of T1 topologies on an infinite set is called T1-independent if their set-theoretic intersection is the cofinite topology, and transversal if their set-theoretic union generates the discrete topology. We show that every Hausdorff space admits a transversal compact Hausdorff topology. Then we apply Booth's Lemma to prove that no infinite set of cardinality less than continuum admits a pair of T1-independent Hausdorff topologies. This answers, in a strong form, a question posed by S. Watson in 1996. It is shown in ZFC that the remainder of the Stone-Cech compactification of the countable discrete set is a self T1-independent compact Hausdorff space, but the existence of self T1-independent compact Hausdorff spaces of cardinality continuum is both consistent with and independent of ZFC.