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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 $\mathbb{R}$?

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Compact subsets of $\beta \mathbb{R}$ containing $\mathbb{R}$, almost by definition. Whether this counts as a classification, of course, depends on what you mean by "classification." – Qiaochu Yuan Nov 17 '12 at 9:15
A very nice answer would be a set of explicit topological spaces (as the examples given above), but it depends on how nice are the compactifications of $\mathbb{R}$. – Seirios Nov 17 '12 at 9:37
@Qiaochu: That’s some but not all: just as $\omega+1$ isn’t a subspace of $\beta\omega$, $S^1$ isn’t a subspace of $\beta\Bbb R$. What is true is that if $c\Bbb R$ is a compactification of $\Bbb R$, there is a continuous surjection $f:\beta\Bbb R\to c\Bbb R$ whose restriction to $\Bbb R$ is the identity. – Brian M. Scott Nov 17 '12 at 9:38
@Brian: my apologies, I got confused about the direction of the universal property. – Qiaochu Yuan Nov 17 '12 at 9:41
@Qiaochu: No problem; life would be simpler in many ways if it did go the other way. (Though Anatole Beck would have denied this strenuously when I was in grad school, pointing out that mathematics would then be inconsistent!) – Brian M. Scott Nov 17 '12 at 9:44
up vote 1 down vote accepted

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 git 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.

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In some sense, you answered my question; the compactifications of $\mathbb{R}$ are not as nice as I expected. – Seirios Nov 24 '12 at 11:07

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