closed and discrete subset of a metric space

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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.






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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?
    – Bungo
    Sep 2 at 6:21















up vote
0
down vote

favorite












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



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






metric-spaces







share|cite|improve this question

















  • 1




    How about $1, 1/2, 1/3, 1/4, ldots$ in the space $mathbb R setminus 0$ with the usual metric?
    – Bungo
    Sep 2 at 6:21













up vote
0
down vote

favorite









up vote
0
down vote

favorite











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



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






metric-spaces







share|cite|improve this question













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



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






metric-spaces




metric-spaces






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Sep 2 at 6:12









Jave

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  • 1




    How about $1, 1/2, 1/3, 1/4, ldots$ in the space $mathbb R setminus 0$ with the usual metric?
    – Bungo
    Sep 2 at 6:21













  • 1




    How about $1, 1/2, 1/3, 1/4, ldots$ in the space $mathbb R setminus 0$ with the usual metric?
    – Bungo
    Sep 2 at 6:21








1




1




How about $1, 1/2, 1/3, 1/4, ldots$ in the space $mathbb R setminus 0$ with the usual metric?
– Bungo
Sep 2 at 6:21





How about $1, 1/2, 1/3, 1/4, ldots$ in the space $mathbb R setminus 0$ with the usual metric?
– Bungo
Sep 2 at 6:21











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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.






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    up vote
    0
    down vote



    accepted










    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.






    share|cite|improve this answer
























      up vote
      0
      down vote



      accepted










      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.






      share|cite|improve this answer






















        up vote
        0
        down vote



        accepted







        up vote
        0
        down vote



        accepted






        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.






        share|cite|improve this answer












        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.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Sep 2 at 10:24









        Henno Brandsma

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