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We know that metric spaces are normal. We also know that a uniform space is Hausdorff if the intersection of all entourages is the diagonal, in which case it is even regular.

However, is there a necessary and sufficient criterion to ensure that a uniform space is normal (A sufficient condition would be, for example, the space is metrizable)? I have read Bourbaki, General Topology as well as Willard, General topology, but I was not able to find any.

Thanks in advance.

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    $\begingroup$ Interesting question, I think. You might be aware that uniform spacs also have a cover reformulation, and normality does have a cover equivalent (point finite covers are shrinkable). This might be the way to go, I'm not sure. I at least never heard of such a charcterisation, but then again, I'm no expert on uniform spaces. $\endgroup$ Commented Feb 19, 2017 at 21:34
  • $\begingroup$ Thank you very much. I will check it~ @HennoBrandsma $\endgroup$
    – Ran Wang
    Commented Feb 19, 2017 at 23:18

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