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.

  • 1
    $\begingroup$ Uniform spaces allow you to discuss uniform continuity in more general settings than metric spaces, for instance topological groups always have a compatible uniform structure. Thus you can prove results like "any continuous map is uniformly continuous on a compact uniform space" in this setting, which can turn out to be interesting; for instance in topological dynamics $\endgroup$ – Max Mar 27 at 11:29
  • $\begingroup$ Thanks for the suggestion. Do you have specific examples or theorems I should look at? $\endgroup$ – jessica Mar 27 at 12:44
  • $\begingroup$ The Hahn Banach theorem is a statement about vector spaces; an algebraic structure is needed to discuss the theorem. Also, uniform spaces do provide generalizations of some results from topological group/vector space theory as you touch in your post. For instance: a topological space is uniformalizable iff it’s completely regular; every topological group (and hence topological vector space) can be endowed with a natural uniform structure, so one sees that topological groups are completely regular. $\endgroup$ – LinearOperator32 Apr 1 at 8:49
  • $\begingroup$ The notion of uniform spaces is also important in commutative ring theory. Given a maximal ideal $M$ of a commutative ring $R$, we can define the $M$-adic uniformity on $R$ which has many applications. For instance, if $R$ is an integral domain, we can characterize the prime spectrum of the ring $Int(R)$ of integer-valued polynomials, using that these polynomials are $M$-adically uniformly continuous. $\endgroup$ – Daniel W. Aug 19 at 20:16

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