We deal with a conjectured dichotomy for compact Hausdorff spaces: each such space contains a non-trivial converging ω-sequence or a non-trivial converging ω₁-sequence. We establish that this dichotomy holds in a variety of models; these include the Cohen models, the random real models and any model obtained from a model of CH by an iteration of property K posets. In fact in these models every compact Hausdorff space without non-trivial converging ω₁-sequences is first-countable and, in addition, has many ℵ₁-sized Lindelöf subspaces. As a corollary we find that in these models all compact Hausdorff spaces with a small diagonal are metrizable.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm224-3-1,
author = {Alan Dow and Klaas Pieter Hart},
title = {Reflecting Lindelof and converging o1-sequences},
journal = {Fundamenta Mathematicae},
volume = {227},
year = {2014},
pages = {205-218},
zbl = {1297.54050},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm224-3-1}
}
Alan Dow; Klaas Pieter Hart. Reflecting Lindelöf and converging ω₁-sequences. Fundamenta Mathematicae, Tome 227 (2014) pp. 205-218. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm224-3-1/