British Rail Express Metric (SNCF Metrica)

British Rail Express Metric

Sia $X={\mathbb{R}}^2$ e $\delta :{\mathbb{R}}^2\to R$ definita come:
$\delta (x,y)=\{\begin{array}{ll}\underline{0}& \text{se }x=y\\ d(x,\underline{0})+y(y,\underline{0})& \text{se }x\ne y\end{array}$
Si dimostra che soddisfa le proprietà di metrica.
Le palle aperte di $\delta$ sono del tipo:
$B_d(x,r)=\{\begin{array}{ll}{B}_d(\underline{0},r)& \text{se }x=\underline{0}\\ \{x\}& \text{se }x\ne \underline{0}\wedge r<d(x,\underline{0})\\ \{x\}\cup B_d(\underline{0},r-d(x,\underline{0}))& \text{se }x\ne \underline{0}\wedge r\ge d(x,\underline{0})\end{array}$