MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

This question already has an answer here:

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?

share|cite|improve this question

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

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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

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.

share|cite|improve this answer
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

Not the answer you're looking for? Browse other questions tagged or ask your own question.