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?

Thanks ahead.

  • $\begingroup$ 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? $\endgroup$ – David White Mar 18 '13 at 1:11
  • $\begingroup$ It may be not at all. $\endgroup$ – Paul Mar 18 '13 at 1:45

Please cheek 5I in gilman jerison Book (Rings of continuous functions). I am not sure. Since I dont put comments, I put this as an answer.

  • $\begingroup$ What is meaning of "5l"? $\endgroup$ – Paul Mar 18 '13 at 1:42
  • $\begingroup$ Exercise 5I in Book of Gilman jerison. It is a completely regular, pseduocompact, non-normal, satisfies the first countably axiom, and every subset of this space is $G_{\delta}. But is not contably compact. Because, has an infinite discrete zero-set. $\endgroup$ – Ali Mar 18 '13 at 1:48
  • $\begingroup$ +1: Hopefully you can leave comments soon $\endgroup$ – David White Mar 18 '13 at 2:03
  • $\begingroup$ @Ali: could you give me a link of the book, so I can download it? $\endgroup$ – Paul Mar 18 '13 at 2:10
  • 1
    $\begingroup$ Ok, you can downloaded this book of BookFinder and search ring of continuous fuctions. $\endgroup$ – Ali Mar 18 '13 at 2:17

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