Sequential compactness vs. countable compactness
Volume 120 / 2010
The general question of when a countably compact topological space is sequentially compact, or has a nontrivial convergent sequence, is studied from the viewpoint of basic cardinal invariants and small uncountable cardinals. It is shown that the small uncountable cardinal $\frak h$ is both the least cardinality and the least net weight of a countably compact space that is not sequentially compact, and that it is also the least hereditary Lindelöf degree in most published models. Similar results, some definitive, are given for many other cardinal invariants. Special attention is paid to compact spaces. It is also shown that MA$(\omega_1)$ for $\sigma$-centered posets is equivalent to every countably compact $T_1$ space with an $\omega$-in-countable base being second countable, and also to every compact $T_1$ space with such a base being sequential. No separation axioms are assumed unless explicitly stated.