Sequential compactness vs. countable compactness

Angelo Bella¹ ; Peter Nyikos²

¹ Dipartimento di Matematica viale A. Doria 6 95125 Catania, Italy
² Department of Mathematics University of South Carolina Columbia, SC 29208, U.S.A.

Colloquium Mathematicum, Tome 120 (2010) no. 2, pp. 165-189

Résumé

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.

Détail

DOI : 10.4064/cm120-2-1

Keywords: general question countably compact topological space sequentially compact has nontrivial convergent sequence studied viewpoint basic cardinal invariants small uncountable cardinals shown small uncountable cardinal frak least cardinality least net weight countably compact space sequentially compact least hereditary lindel degree published models similar results definitive given many other cardinal invariants special attention paid compact spaces shown omega sigma centered posets equivalent every countably compact space omega in countable base being second countable every compact space base being sequential separation axioms assumed unless explicitly stated