On countable compactness and sequential compactness

Zhou, Hao Xuan

doi:10.1090/S0002-9939-1984-0722428-3

On countable compactness and sequential compactness
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Proc. Amer. Math. Soc. 90 (1984), 121-127 Request permission

Abstract:

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

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