I recently heard a lecture where the following fact was mentioned without further comment:
The space $\omega_1$ of all countable ordinals is not completely uniformizable. (I.e., there is no complete uniformity on $\omega_1$ which induces the usual (order) topology on $\omega_1$.)
This seemed interesting, and I thought (naively and incorrectly, as it turns out) that it wouldn't be too difficult to prove. Honestly, I haven't got anywhere with it. Virtually all of my knowledge of uniform spaces comes from I.M. James' "Topologies and Uniformities".
I'm hoping that someone here can at least outline a proof of this fact. Thanks in advance!