Sequential compactness vs. countable compactness

Angelo Bella, and Peter Nyikos. "Sequential compactness vs. countable compactness." Colloquium Mathematicae 120.2 (2010): 165-189. <http://eudml.org/doc/284225>.

@article{AngeloBella2010,
abstract = {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 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(ω₁) for σ-centered posets is equivalent to every countably compact T₁ space with an ω-in-countable base being second countable, and also to every compact T₁ space with such a base being sequential. No separation axioms are assumed unless explicitly stated.},
author = {Angelo Bella, Peter Nyikos},
journal = {Colloquium Mathematicae},
keywords = {countably compact; compact; sequentially compact; KC space; cardinal invariants; splitting tree; network; weight; net weight; hereditary Lindelöf degree; Novak number; -in-countable base; ponderous},
language = {eng},
number = {2},
pages = {165-189},
title = {Sequential compactness vs. countable compactness},
url = {http://eudml.org/doc/284225},
volume = {120},
year = {2010},
}

TY - JOUR
AU - Angelo Bella
AU - Peter Nyikos
TI - Sequential compactness vs. countable compactness
JO - Colloquium Mathematicae
PY - 2010
VL - 120
IS - 2
SP - 165
EP - 189
AB - 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 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(ω₁) for σ-centered posets is equivalent to every countably compact T₁ space with an ω-in-countable base being second countable, and also to every compact T₁ space with such a base being sequential. No separation axioms are assumed unless explicitly stated.
LA - eng
KW - countably compact; compact; sequentially compact; KC space; cardinal invariants; splitting tree; network; weight; net weight; hereditary Lindelöf degree; Novak number; -in-countable base; ponderous
UR - http://eudml.org/doc/284225
ER -