# non-trivial convergent sequence [duplicate]

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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## marked as duplicate by Bill Johnson, Joseph Van Name, Daniel Moskovich, Chris Godsil, David WhiteAug 25 '13 at 14:15

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
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.
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