On countable compactness and sequential compactness

Abstract

If a countably compact T 3 {T_3} space X X can be expressed as a union of less then c c many first countable subspaces, then MA implies that X X is sequentially compact. Also MA implies that every countably compact space of size > c > c is sequentially compact. However, there is a model of ZFC in which ω 1 > c {\omega _1} > c and there is a countably compact, separable T 2 {T_2} space of size ω 1 {\omega _1} , which is not sequentially compact.