  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.
