Whilst reading about extensions of C*-alegbras, this topological fact was stated. I understand why $\beta \mathbb{R} \setminus \mathbb{R}$ has at least 2 connected components (it surjects onto the two point compactification $[-\infty,\infty]$ mapping remainders to remainders) but can't see how to show that there are exactly 2.

The literature on this subset suggests that itself enough to understand the Stone-Cech remainder, $\beta H \setminus H$, of the half line $H = [0,\infty]$. Unfortunately, I've found that the papers on this subject consider the connectedness of this space too basic (for the intended audience) to prove explicitly or provide a reference to a proof.


closed as off-topic by Bjørn Kjos-Hanssen, Matthew Kahle, YCor, Yemon Choi, Joseph Van Name Apr 7 '14 at 19:52

  • This question does not appear to be about research level mathematics within the scope defined in the help center.
If this question can be reworded to fit the rules in the help center, please edit the question.