On hereditarily normal topological groups

Raushan Z. Buzyakova¹

¹

Fundamenta Mathematicae, Tome 219 (2012) no. 3, pp. 245-251

Résumé

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.

Détail

DOI : 10.4064/fm219-3-3

Keywords: investigate hereditarily normal topological groups their subspaces prove every compact subspace hereditarily normal topological group metrizable prove statement first hereditarily normal topological group non trivial convergent sequence has delta diagonal implies particular every countably compact subspace hereditarily normal topological group non trivial convergent sequence metrizable another corollary under proper forcing axiom every countably compact subspace hereditarily normal topological group metrizable