<s>[INST] Big an open ball inside small open ball in metric space When i was reading the book "Elements of the Theory of Functions and Functional Analysis" (A. N. Kolmogorov, S. V. Fomin), I encountered a very interesting (for me) problem.
Problem:

Create a metric space with two open balls $B(x,\rho_1), B(y,\rho_2)$ such that $\rho_1>\rho_2$ and $B(x,\rho_1)\subset B(y,\rho_2)$.

I will post my example, and i would like to know more examples.
Thanks for attention!
 [/INST] One example: 
metric space $(X,d)$ such that $X(0,1)$ with metrica $d\left( x,y \right) =\begin{cases} 2-\left| x-y \right| \quad ,x\neq y \\ 0\quad \quad \quad \quad \quad \quad ,x=y \end{cases}$ 
$$d\left( \frac { 1 }{ 2 } ,\frac { 1 }{ 2 } +\frac { 1 }{ 5 }  \right) =2-\frac { 1 }{ 5 } =\frac { 9 }{ 5 } >\frac { 7 }{ 4 } =2-\frac { 1 }{ 4 } =d\left( \frac { 1 }{ 2 } ,\frac { 1 }{ 2 } +\frac { 1 }{ 4 }  \right) \\$$
however $$\left( \frac { 1 }{ 2 } -\frac { 1 }{ 5 } ,\frac { 1 }{ 2 } +\frac { 1 }{ 5 }  \right) \subset \left( \frac { 1 }{ 2 } -\frac { 1 }{ 4 } ,\frac { 1 }{ 2 } +\frac { 1 }{ 4 }  \right) $$
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