Reflecting Lindelöf and converging $\omega _{1}$-sequences
Volume 224 / 2014
Fundamenta Mathematicae 224 (2014), 205-218
MSC: Primary 54D30; Secondary 03E35, 54A20, 54A25, 54A35, 54D20.
DOI: 10.4064/fm224-3-1
Abstract
We deal with a conjectured dichotomy for compact Hausdorff spaces: each such space contains a non-trivial converging $\omega $-sequence or a non-trivial converging $\omega _1$-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 $\mathsf {CH}$ by an iteration of property K posets. In fact in these models every compact Hausdorff space without non-trivial converging $\omega _1$-sequences is first-countable and, in addition, has many $\aleph _1$-sized Lindelöf subspaces. As a corollary we find that in these models all compact Hausdorff spaces with a small diagonal are metrizable.
