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Consider a uniform space $X$ (with induced topology).

What of the following can be a subset of the other? Which of the following is always a subset of the other?

  1. Neighborhood of the diagonal in the topology corresponding to $X\times X$.
  2. The set of entourages of our uniformity.
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    $\begingroup$ "...with induced topology"....Induced from where or what? $\endgroup$ – DonAntonio Sep 22 '13 at 21:18
  • $\begingroup$ Induced by the uniform structure, presumably. $\endgroup$ – Daniel Fischer Sep 22 '13 at 21:19
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    $\begingroup$ The entourages are always neighbourhoods of the diagonal in the induced topology. I'll leave finding the proof to you. $\endgroup$ – Daniel Fischer Sep 22 '13 at 21:20
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As Daniel Fisher wrote in a comment, every entourage is a neighborhood of the diagonal. The converse fails, for example, if $X$ is the real line with the uniform structure induced by the usual metric. Specifically, $\{(x,y): y<x+e^{-x}\}$ is an open neighborhood of the diagonal, but not an entourage (because its upper boundary gets too close to the diagonal for large $x$).

If I remember correctly, though, a compact Hausdorff space always has a unique uniform structure, and the entourages of this structure are exactly the neighborhoods of the diagonal.

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  • $\begingroup$ Yes, you remember correctly, this is Theorem 8.3.13 from Engelking’s “General Topology”. $\endgroup$ – Alex Ravsky Apr 21 at 9:37

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