Give an example of metric space that contains balls $B(x_1,r_1)\subsetneqq B(x_2,r_2)$, with $r_1>r_2$.

Was initially thinking about discrete metric, however, in discrete case one can never achieve a proper subset with given condition.


In the metric on $\{0,1,2\}$ induced by the standard metric on $\mathbb R$, $B(0,1.2)\subsetneq B(1,1.1)$.

  • $\begingroup$ How did you come up with that? :< $\endgroup$ – Alvin Lepik Mar 22 '16 at 17:46
  • 2
    $\begingroup$ @AlvinLepik: That question is much harder than the original question :-) I drew a picture of one circle containing another and wondered how their "centres" would have to lie for this to work out; that suggested that the "centre" of the inner circle would have to be "off-centre" and the outer circle would have to contain a point further away from it than from the outer circle's "centre". $\endgroup$ – joriki Mar 22 '16 at 17:50

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