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Is every closed and discrete subset of a metric space uniformly discrete?

I tried searching for a counterexample but could not find any.

Jave
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    How about ${1, 1/2, 1/3, 1/4, \ldots}$ in the space $\mathbb R \setminus {0}$ with the usual metric? –  Sep 02 '18 at 06:21

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Let $X$ be the metric space $(0,1]$ in the usual metric.

Then $A = \{\frac1n: n =1,2,3,\ldots\}$ is closed and discrete but not uniformly so.

Henno Brandsma
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