Metric-space with a ball inside a smaller ball [closed]

Question

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 $B(x_2,r_2)\subset B(x_1,r_1)$?

Answer 1

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.

Answer 2

Take $X$ to be the closed first quadrant in the plane, $\rho$ the restriction of the Euclidean metric to $X$, $x_2=(0,0)$, $r_2=2$, $x_1=(1,1),\; r_1=\sqrt{2}$. The balls are closed. By slightly changing $r_1,r_2$ you can make them open.

Answer 3

The easiest example is $X = \{x\}$.