This title might be slightly misleading since I saw that the name 'uniformization' theorem is already used in a different context, but my question regards whether a topological space arises from a uniformity.
I know that there is a metrization theorem characterizing fully a metrizable topological space. I was wondering whether there is a similar result regarding if a topological space is 'uniformisable', even partially. I have very little acquaintace with uniform spaces, so I would appreciate even simple somewhat 'obvious' answers.
Yes, such a theorem exists: the topology of a topological space $X$ is induced by a uniform structure if and only $X$ is completely regular; in other words, if it is a $T_{3\frac12}$ space.