Classification of the compactifications of $\mathbb{R}$ 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 compactification 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 $\mathbb{R}$?
 A: This isn’t an answer; it’s more an indication of why a nice answer isn’t likely to be forthcoming, at least in purely topological terms.
Since there’s a nice bijection between compactifications of $\Bbb R$ and algebras $\mathscr{A}\subseteq C^*(\Bbb R)$ that separate points and closed sets and are closed in the sup norm, you can turn the problem into one of classifying these algebras; more might be known from that point of view.
This paper mentions a classification of a large subset of the compactifications that is also algebraic in nature, but a bit easier to get a grip on: in Section 8.4 that the lattice of all topological group compactifications of $\Bbb R$ is isomorphic to the lattice of subgroups of $\Bbb R_d$, the discretization of $\Bbb R$, ordered by $\subseteq$. For something more topological, this paper isn’t an attempt at a large-scale classification, but it does construct a class of compactifications of $\Bbb R$ that includes some that (unlike $\beta\Bbb R$) can be visualized, if not quite so easily as the one- and two-point compactifications.
