Reflecting Lindelöf and converging $\omega _{1}$-sequences

Alan Dow ¹   ; Klaas Pieter Hart ²

¹ Department of Mathematics UNC-Charlotte 9201 University City Blvd. Charlotte, NC 28223-0001, U.S.A.
² Faculty of Electrical Engineering, Mathematics and Computer Science TU Delft Postbus 5031 2600 GA Delft, the Netherlands

Fundamenta Mathematicae, Tome 224 (2014) no. 3, pp. 205-218

Résumé

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.

Détail

DOI : 10.4064/fm224-3-1

Keywords: conjectured dichotomy compact hausdorff spaces each space contains non trivial converging omega sequence non trivial converging omega sequence establish dichotomy holds variety models these include cohen models random real models model obtained model mathsf iteration property posets these models every compact hausdorff space without non trivial converging omega sequences first countable addition has many aleph sized lindel subspaces corollary these models compact hausdorff spaces small diagonal metrizable