non trivial convergent sequence

Let $\beta\omega$ be the Stone-Čech compactification of the natural numbers. We know that it is compact and Hausdorff, but it has no non-trivial convergent sequence.

Is there an example else to be compact and Hausdorff, but it has no non-trivial convergent sequence ?

• Probably not what you are looking for, but any finite space, with the discrete topology has this property... – N. S. Aug 14 '13 at 17:49
• A finite discrete space has only eventually constant convergent sequences. But I think you want something more sophisticated. – Stefan Hamcke Aug 14 '13 at 17:49
• I think in a Hausdorff space this is equivalent to each subset being sequentially closed. So the space cannot be sequential if you don't want the discrete topology. – Stefan Hamcke Aug 14 '13 at 17:55