Could you tell me an example to an $(X,\varrho)$ metric-space with balls $B(x\_1,r\_1)$ and $B(x\_2,r\_2)$ where $r\_1<r\_2$ but also $B(x\_2,r\_2)\subset B(x\_1,r\_1)$?



Metric-space with a ball inside a smaller ball
A slightly simpler, 1-dimensional version of Alexandre's example: On the closed half-line $[0,\infty)$, the ball of radius 3 around 0 (i.e., the interval $[0,3)$) equals the ball of radius 2 around the point 1. For a proper inclusion of balls, shrink 3 or enlarge 2 slightly.


