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Compact groups of Ulam-measurable cardinality: Partial converses to theorems of Arhangel’skiĭ and Varopoulos. (English) Zbl 0817.22006
By a theorem of A. V. Archangel’skij [Topol. Appl. 57, 163-181 (1994; Zbl 0804.54001)], if a compact topological group $$G$$ admits a strictly finer countably compact topological group topology, then $$| G|$$ is an Ulam-measurable cardinal. The authors prove a counterpart of that theorem: If $$G$$ is a compact group with $$| G|$$ Ulam- measurable (and $$G$$ is Abelian, or connected), then $$G$$ admits a strictly finer countably compact topological group topology. Furthermore, there are a compact group $$K$$ and a sequentially continuous homomorphism of $$G$$ to $$K$$ which is discontinuous.

MSC:
 22C05 Compact groups 03E55 Large cardinals 54A35 Consistency and independence results in general topology 54H11 Topological groups (topological aspects) 54A10 Several topologies on one set (change of topology, comparison of topologies, lattices of topologies)