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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.

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  • $\begingroup$ Why $[0,1]^I$ has the FPP? $\endgroup$ – G. Ottaviano Aug 16 '18 at 22:40
  • $\begingroup$ @G.Ottaviano It's a classic well-known fact. It extends Brouwer's fp theorem for finite $I$. $\endgroup$ – Henno Brandsma Aug 16 '18 at 22:48
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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

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