On hereditarily normal topological groups

Volume 219 / 2012

Raushan Z. Buzyakova Fundamenta Mathematicae 219 (2012), 245-251 MSC: Primary 54H11, 22A05, 54D15. DOI: 10.4064/fm219-3-3

Abstract

We investigate hereditarily normal topological groups and their subspaces. We prove that every compact subspace of a hereditarily normal topological group is metrizable. To prove this statement we first show that a hereditarily normal topological group with a non-trivial convergent sequence has $G_\delta $-diagonal. This implies, in particular, that every countably compact subspace of a hereditarily normal topological group with a non-trivial convergent sequence is metrizable. Another corollary is that under the Proper Forcing Axiom, every countably compact subspace of a hereditarily normal topological group is metrizable.

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