Samuel compactification of the real line Is there any example of metric t over the real line R, compatible with the Euclidean topology, satisfying that (R, t) is a complete metric space but its Samuel compactification is not homeomorphic to the Samuel compactification of (R,d), where d denotes the usual Euclidean metric?
There are several ways to define the Samuel compactification of a metric space (or uniform space). I recommend to give a look to the paper of R. G. Woods, The minimum uniform compactification of a metric space, to have an idea. In the frame of topological groups it is sometimes called the greatest ambit.
The definition  I work with is the following: the Samuel compactification of a metric space (X,d) is the completion of (X, $f\mu _d$) where $f\mu _d$ denotes the uniformity having as a base all the finite uniform covers from the uniformity $\mu _d$ induced by the metric (the uniformity $f\mu _d$ is always compatible with the topology of X).
Thus,  (R,t) and (R,d) have homeomorphic Samuel compactifications if and only if there is an hemeomorphism $\varphi: ({\bf R},f\mu _t) \rightarrow ({\bf R} ,f\mu _d) $ such that  $\varphi$ and $\varphi ^{-1}$ preserves Cauchy filters.
Definition: A function between two uniform spaces it said to preserve Cauchy filters if and only it sends Cauchy filters to Cauchy filters.
 A: I don't know the exact answer to your question, but this may help:
I usually prefer using Smirnov compactifications instead. I believe that Samuel compactification was first defined for uniform spaces, whereas Smirnov compactification was defined for proximity spaces. However, since uniform spaces and proximity spaces have a very close relationship, the Samuel compactification of a metric space with its metric uniformity is homeomorphic to the Smirnov compactification of a metric space with its metric proximity. So let us focus on Smirnov compactifications and proximities for now.
Proximities have associated induced topologies and compactifications. It is possible for two different proximities to induce the same topology on the original space, but to have different compactifications. For example, given the real line, you can define 2 proximities:
1) $A$ close to $B$ iff the distance between $A$ and $B$ is $0,$
2) $A$ close to $B$ iff their closures intersect or they are both unbounded
Both of these proximities induce the Euclidean topology on the real line. However, the Smirnov compactification of the real line coming from $1)$ is the minimum uniform compactification, whereas the Smirnov compactification of the real line coming from $2)$ is the $1$-point compactification.
In fact, there is a bijective correspondence between proximites compatible with the given topology and compactifications of that space. Thus, to positively answer your question, you would need to find two different proximity structures that are compatible with the Euclidean metric (i.e., they induce the Euclidean metric). I believe that it is possible, since I have seen a result stating that the Stone Cech compactification can be realized as a supremum of Smirnov compactifications coming from metric proximities.
I hope that you find these remarks useful/interesting.
A: I realize I'm answering a very old question, but in the interest of completeness, according to this and that answers to the same question on MathOverflow, the Samuel compactification of $\mathbb{R}$ can be identified with the disjoint union of $\beta\mathbb{Z}$ and $\beta\mathbb{Z} \times (0,1)$ where $\beta\mathbb{Z}$ is the Stone-Čech compactification of $\mathbb{Z}$.
