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Can you recommend some literature that give a proof of this statement, and who allegedly prove it first?

BTW, is there any use of uniform spaces or topological spaces in mathematical (or theoretical) physics?

Thanks.

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I think you can find this in Kelley's General topology. – Benjamin Steinberg Nov 17 '11 at 20:57
    
It is in the general topology book of Bourbaki. If that is not the first published proof, it is probably in the paper of Andre Weil that introduced uniform space as entourages. – Michael Greinecker Nov 17 '11 at 21:27

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