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I have reached a deadlock to find a example to show that a compact Hausdorff space does not need to have a no non-trivial convergent sequence.(except $\beta\omega$)

can you give me a example of a compact Hausdorff space with no non-trivial convergent sequence?

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In a comment on the previous question I stated that every compact $F$-space is an example of a compact Hausdorff space with no non-trivial convergent sequence. Please do not repeat a question that has been answered already. –  Joseph Van Name Aug 25 '13 at 12:24
    
A one-point space? –  Gerald Edgar Aug 25 '13 at 12:54
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marked as duplicate by Bill Johnson, Joseph Van Name, Daniel Moskovich, Chris Godsil, David White Aug 25 '13 at 14:15

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As far as I know, the only known example is $\beta\omega$ (and spaces that contain copies of $\beta\omega$), I sort of recall (but don't take my word on that) that the question of whether there exists other example is still open.

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There are consistent examples of infinite compact spaces which contain no non-trivial convergent sequence and no copy of $\beta\omega$. It is still open whether there is an example in $ZFC$. –  Ramiro de la Vega Aug 25 '13 at 23:44
    
I see, so the point is getting the thingy in $\mathscr{ZFC}$, or proving that it consistently doesn't exist. Do you have any idea in which models do those examples live? –  David FernandezBreton Sep 24 '13 at 16:08
    
The classic examples are due to Fedorchuk; one is constructed under $\diamondsuit$, another under $CH$ and a third one under $\mathfrak{s}=\aleph_1 + 2^{\aleph_0}=2^{\aleph_1}$. For more details, search for "Efimov's problem". –  Ramiro de la Vega Sep 24 '13 at 23:47
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