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 -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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm219-3-3,
title = {On hereditarily normal topological groups},
journal = {Fundamenta Mathematicae},
volume = {219},
year = {2012},
pages = {245-251},
zbl = {1270.54037},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm219-3-3}
}
(éd.). On hereditarily normal topological groups. Fundamenta Mathematicae, Tome 219 (2012) pp. 245-251. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm219-3-3/