Reflecting Lindel\"of and converging omega_1-sequences

TL;DR

This paper investigates a conjecture about the structure of compact Hausdorff spaces, showing it holds in various models and linking the absence of certain sequences to metrizability.

Contribution

It proves the conjectured dichotomy in multiple models and connects the absence of omega_1-sequences to first-countability and metrizability.

Findings

The dichotomy holds in Cohen, random real, and certain iterated models.

Spaces without omega_1-sequences are first-countable and have many Lindelöf subspaces.

Compact spaces with a small diagonal are metrizable in these models.

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 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 Lindel\"of subspaces. As a corollary we find that in these models all compact Hausdorff spaces with a small diagonal are metrizable.