Why are uniform spaces important? I've thought of two possible good answers:
Topological groups. According to nCat locally compact groups are complete with respect to the left/right uniformity, and the Haar measure is defined on locally compact groups.
Functional analysis. Every seminorm induces a pseudometric, and I've read that uniform spaces can be defined via pseudometrics. This suggests to me that important results like the Hahn-Banach theorem would hold for complete uniform spaces. Is this true?
Unfortunately I lack the background to understand the Haar measure properly, and I'm only familiar with the entourage version of uniform spaces.
Any exposition and thoughts on what I've written above would be highly appreciated! Other important consequences of uniform structures would be great too.