# The Jungle River metric induced topology compactness

I have the Jungle River metric induced topology, $\tau$, given by the metric:

$$d(x,y) = \begin{cases} |x_2-y_2|, & \text{if x_1 = y_1;} \\ |x_2| + |y_2| + |x_1-y_1|, & \text{if x_1 \neq y_1 } \end{cases}$$

I need to prove that the topological space $(\mathbb{R}^2,\tau)$ is not compact.

Let $x=(x_1,x_2)$ then the balls are:

$$B_d=\{x_1\}\cdot(x_2-\varepsilon,x_2+\varepsilon) \cup B_1((x_1,0), \varepsilon-|x_2|)$$

where $B_1$ is the the ball with the $d_1$ metric.

I have problems finding a cover for $\mathbb{R}^2$.