TITLE: Stone–Čech compactification of $\mathbb{N}, \mathbb{Q}$ and $\mathbb{R}$
QUESTION [12 upvotes]: I'm trying to find connections between Stone–Čech compactifications of $\mathbb{N}, \mathbb{Q}$ and $\mathbb{R}$, all with the euclidean topology. So, are there any ? e.g. is $\beta \mathbb{Q} = \beta \mathbb{R}$ ? I know that the cardinality of $\beta \mathbb{N} = \beta \mathbb{Q} =  \beta \mathbb{R}$ thanks to Wikipedia, but how does one prove it ?

REPLY [8 votes]: In answer to the first part of your question, it's not true that $\beta\mathbb Q=\beta\mathbb R$. For suppose $g$ is some autohomeomorphism of $\mathbb R$. This extends to an autohomeomorphism $\beta g$ of $\beta\mathbb R$, and if we had $\beta\mathbb  Q=\beta\mathbb R$, it would follow that $\beta g$ is an autohomeomorphism of $\beta\mathbb Q$. Since the restriction of $\beta g$ to $\mathbb Q$ is then an autohomeomorphism of $\mathbb Q$ (see the paper cited below), it would follow that every autohomeomorphism of $\mathbb R$ maps $\mathbb Q$ to itself, and this is clearly not true (e.g., $x \mapsto x +\sqrt2$).
Re. comparison of the compactifications, it's potentially worth looking at the remainders. It turns out, for example, that $\mathbb Q$* is separable but $\mathbb R$* isn't. See: M. P. Stannett, "The Stone-Čech Compactification of the Rational World", Glasgow Math. J., vol. 30 (1988) pp. 181-188.