Is the long line completely uniformizable? The long line $L$ is uniformizable; in fact, as $L$ is a locally compact Hausdorff space we can explicitly write down a uniformity for it: If $\hat{L}$ is the one-point compactification of $L$, then $\hat{L}$ is compact Hausdorff, and so it has a natural uniformity that we can restrict to $L$.
But this is not a complete uniformity; its completion is $\hat{L}$.  Is there a complete uniformity on $L$?
One well-known result is that any paracompact Hausdorff space is completely uniformizable.  In particular, any manifold is completely uniformizable.  Does the same apply to spaces like $L$, or are they "too noncompact" to have a complete uniformity?
 A: There is only one compatible uniformity with the topology on $L$. This follows from this paper  or the well-known result (reference anyone?) that locally compact Tychonoff spaces have a unique uniformity iff the one-point compactification is the same as the Cech-Stone compactification. So $L$ has a unique uniformity that is totally bounded and not complete. 
Added: I misinterpreted $L$ as being the long ray $R$ (which is the space  $\omega_1 \times [0,1)$, in the lexicographic order and order topology) and then the result holds (the unicity). The long line is two copies of the long ray, one ordered reversely, the other as above, glued together at their endpoints $(0,0)$. 
$L$ has two compactifications: the ordered compactification, which adds two points (one at $+\omega_1$ the other $-\omega_1$, as it were), and the one-point compactification of $L$. The former also equals the Cech-Stone compactification $\beta(L)$. This is by the well-known fact that every real-valued function on the long ray is eventually constant. The constants can differ between the left and the right side, and so the classification in the previous paper (the last theorem) shows that $L$ has at least two compatible uniformities (e.g. there are continuous functions $f$ to the reals such that neither $\{x: f(x) \le 0 \}$ and $\{x: f(x) \ge 1\}$ are compact). (The Cech-Stone compactification for $L^n$ is $\beta(L)^n$ because of Glicksberg's theorem (as $L$ is pseudo-compact and locally compact)).
So this still leaves open the question whether $L$ has a compatible complete uniformity (my guess is still no). Does it have exactly two (my guess is yes)?
