Are there any important examples of uniform spaces other than metric spaces and topological groups?
Also, what is an example of when the uniform structure of topological groups is used?
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A simple example: $[0,1]^I$ is a uniform space (as a product of metric, hence uniform, spaces). And it's not metrisable if $I$ is uncountable and it's not a topological group as it has the FPP.
In fact, any Tychonov space that is non-metrisable and non-homogeneous is an example.
The uniform structure on topological groups is used e.g. for studying completions, and Haar measure as well.
Examples arbitrary commact Hausdorff spaces, that have a unique uniformity. This example is very interesting, in that it allows to talk about uniform continuity for maps $X\to C$, where, say $X$ is metric and $C$ is compact.
One way uniform spaces are interesting is in the idea of Cauchy completion, which simply generalizes the one for metric spaces. This allows to associate to a given uniform space another one that has nice properties; and it can help in studying, e.g. the topological dynamics of a certain group