I looked at quite a bit of questions on the site and didn't quite find this but apologies if it's here already.
I was wondering if any one knows if there exists an uncountable space and some Stone–Čech compactification that adds exactly two elements , i.e. $ \lvert \beta X \setminus X \lvert = 2$.
I thought about perhaps starting with $ \Omega_0 = [0, \omega_1) $ and sort of iterating compactifications if that makes sense. First adding $ \omega_1$ to the space which I believe gives us a one point compactification that is the same as the Stone–Čech compactification and then trying to do the same to the space that results from that ( i.e. $\Omega$). The problem is that if we apply the same one point compactification process I believe that since $ \Omega$ is already compact we will end up with a space which has a point isolated. Thus ruling out that the compactification is Stone–Čech .
Perhaps this space is not the one to try and work with. If anyone knows of an example that works or has any ideas that would be appreciated, thanks!