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.