Dear all,

we know, owing to a result appearing in the book of R. M. Dudley (Real analysis and Probability - Theorem 2.8.2), that for any separable metric space (S, d), there is a metric e on S, defining the same topology as d, such that (S, e) is totally bounded.

I would like to know if a uniform continuous function f for the metric d will remain uniform continuous for the constructed metric e.

Regards,

Nathalie

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