I have a question regarding the proof about compactifications of $\mathbb{R}$. I am reading Van Douwen´s paper Characterizations of $\beta \mathbb{Q}$ and $\beta \mathbb{R}$, where he defines:
We call a compactification $\gamma X$ of a space $X$ topological if every autohomeomorphism $h$ of $X$ has a continuous extension $\gamma h: \gamma X \rightarrow \gamma X$.
and then he proves that
for the halfline $\mathbb{H}$, $\alpha \mathbb{H}$ and $\beta \mathbb{H}$ are the only topological compactifications of $\mathbb{H}$.
He then states as an immediate consequence that
$\alpha \mathbb{R}$, $\beta \mathbb{R}$ and the two-point compactification are the only three topological compactifications of $\mathbb{R}$.
My question is: Why is this immediatelly seen? I would like to show this a bit more precisely.
I tried to say this:
For any $x \in \mathbb{R}$, the map $f: x \rightarrow - x$ induces an autohomeomorphism of $\beta \mathbb{R}$ which implies that $\beta[0,\infty)$ is identical with $\beta(-\infty,0]$.
But I am not sure how to conclude exactly the result. Will appreciate any help.
P.S. $\alpha X$ denotes the Alexandroff one-point compactification and $\beta X$ denotes the Stone-Čech compactification.
P.P.S. I am sorry I couldn´t find the paper link online.
