Show that the jungle river (barbed wire) metric is a metric I want to prove that Jungle River metric is indeed a metric space, and determine it is open and closed balls. Firstly, i know that the metric is given by $x,y\in \mathbb{R}^2$, such that $x=(x_1,x_2), y=(y_1,y_2),$ and $$
d(x,y)
= \begin{cases}
|x_2-y_2|, & \text{if } x_1 = y_1,\\
|x_2| + |y_2| + |x_1-y_1|, & \text{if } x_1 \neq y_1.
\end{cases}$$

If $d(x,y)=0,$ then $x_1=x_2$ (why?) and therefore  $|y_1-y_2|=0$ which then yields $y_1=y_2.$ Also, if $x=y,$ then $(x_1,x_2)=(y_1,y_2),$ so that $$d(x,y)=|x_2-y_2|=0.$$
To show that $d(x,y)\leq d(x,z)+d(z,y), z:=(z_1,z_2)$; Consider the following four cases: (a) $x_1=z_1;$ (b) $x_1,x_2,z_1$ are pairwise different; (c) $x_1=x_2$ but $z_1$ is different  (d) $x_1$ is different from $$x_2=z_1.$$
Note that: Drawing some balls in this metric you just seem to get a diamonds of some size along the x-axis and then potentially a vertical line extending from its centre that passes through the tip.  
