The Jungle River metric induced topology compactness

Question

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 $\mathbb{R}^2$.

Answer 1

HINT: Small open balls far away from the "river" are intervals. There are uncountably many of such disjoint, open balls. In particular, the plane with the "river" metric contains the uncountable discrete space, so it cannot be compact.