Topology with a strange distance We define the distance in the plane $\mathbb{R}^2$
$$\rho((x_1,y_1),(x_2,y_2)) =\begin{cases}
|y_1-y_2| & \text{if } x_1=x_2 \\
|y_1|+|x_1-x_2|+|y_2| & \text{if not}
\end{cases}$$
. I want to determine what are the open sets in the topology of $\mathbb{R}^2$ associated with this distance. I'm really stuck with this question, because I don't know how to separate in cases to find the open sets. I think that the points in the $x$-axis must be important, but I don't understand how. 
 A: This is usually called the river metric. In my Dutch intro to metric spaces we called it the "Amazone-metriek" on the plane.
Image the plane really being a "forest" with a river : the $x$-axis, and there are "paths" orthogonal to that river.
If you want to move from $(x,y)$ to $(u,v)$, two things can be the case: you're on the same path, so $x = u$, then we can just move along the path like we're in the reals, and the distance becomes $|y-v|$. If not, we go down to the river (starting from $x$ this is distance $|y|$), then travel to the other point's path via the river so that's distance $|x-u|$ and then go up or down the path to $(u,v)$, distance $|v|$. So we get the distance $|y| + |v| + |x-u|$ in total.
This means that for a point away from the river, say with $y$-coordinate $|y|> 0$, a small ball around it will only be able to "see" vertical points. On the river, we can always see some points close to it in other "paths".
Draw some pictures to build your intuition. It's all about what "small" balls looks like to get a grip on what neighbourhoods of a point are.
A: You should remember that an open set of a topology derived from a metric is a union of open balls. And that if a ball is an element in a topology $T$ then the topology $T_\rho$ is contained in $T$.
