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It is well known that every compact Hausdorff space admits a unique (necessarily complete) uniform structure which is compatible with the topology, and every continuous function from such a space to a uniform space is uniformly continuous.

Are there situations in the general theory of compact Hausdorff spaces where it is beneficial to use the theory of uniform spaces? I can imagine, it might be helpful to prove the existence of certain points by using the convergence of arbitrary Cauchy filters/nets.

One direct application I can think of is, that for compact Hausdorff spaces $X,Y$ and a dense subset $A \subseteq X$ any uniformly continuous map $f : A \to Y$ (w.r.t. to the subspace uniformity on $A$) can be uniquely extended by continuity to a map $\overline{f} : X \to Y$. By characterizing the uniformly continuous maps from $A$ in terms of the topology on $X$ one may obtain an interesting (probably well known) extension result.

Dune
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  • @DanielFischer: Thank you for the suggestion! I forgot, that not every continuous map from $A$ must be uniformly continuous. Now the statement should be correct. – Dune Aug 26 '13 at 10:26
  • @MinimusHeximus: Extendable to which space? A space which contains a proper dense compact Hausdorff subspace cannot be Hausdorff, so that situation seems quite rare. – Dune Aug 27 '13 at 09:56
  • @MinimusHeximus: Yes, that might indeed be helpful. But for this statement we really need the terms of uniform spaces and uniformly continuous maps. I am looking for purely topological results about compact Hausdorff spaces, which can be derived from the theory of uniform spaces. – Dune Aug 27 '13 at 16:56
  • @MinimusHeximus: This statement (which is new to me) is really nice - thank you! Does it become wrong for arbitrary spaces? – Dune Aug 28 '13 at 10:18

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Divisibility:

A compact Hausdorff space is much more than just uniformizible, it is divisible. It's clear that divisibility has much to do with uniformizability. For example every T1 divisible space is uniformizable and so it is T2 and Tychonoff. It's clear how uniform spaces help.

Minimal Uniformity:

A Tychonov space is locally compact iff it allows a compactible minimal uniformity. (according to this thread)

Nets:

If $X$ is a Tychonov space and every net on a dense $A⊆X$ has a convergent subnet, then $X$ is compact. The proof is straightforward in Uniform Space Theory.

There are other connections.

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