# A question on countably compact space

A regular space $X$ is

1. star compact (which implies pseudocompact)
2. with $G_\delta$-diagonal
3. star countable
4. first countable
5. $e(X)\le \aleph_0$ ( in fact it implies star countable)
6. $|X|=\aleph_1$
7. Cech-complete
8. under CH

My question is this: Must $X$ be countably compact?

• I'm sure you already know that if $X$ is normal and pseudocompact then $X$ is countably compact. I don't know enough about these other conditions you list to know whether or not they can get you from regular to normal. I know that paracompactness is enough. Any chance your space $X$ is paracompact? Or that it's locally metrizable? – David White Mar 18 '13 at 1:11