Do you know if there exists a classification of the compactifications of $\mathbb{R}$?
From here, we can deduce that there exist only two compactifications with finite remainder: $[0,1]$ and $\mathbb{S}^1$; and from here, you can show that there doesn't exist a compactication with a countable remainder (but an example is given for a compactification with a remainder of cardinality $\mathfrak{c}$). On the other hand, the biggest compactification of $\mathbb{R}$ is $\beta \mathbb{R}$ with a remainder of cardinality $2^{\mathfrak{c}}$.
Can we deduce a complete classification of the compactifications of $\mathbb{R}$?