This is a question from Metric Spaces by Jain and Ahmad. 
In metric space $(\mathbb R^2, d)$ I need to find if every singleton set is open or not, where
$d(x,y)=|x_1-y_1|+|x_2-y_2|$
for $x=(x_1, x_2)$ and $y=(y_1, y_2)$ in $\mathbb R^2$
My Attempt I know that the open sphere centered at any $x\in \mathbb R^2$ will be an open square and it won't be contained in the singleton set considered, but the answer says that every singleton would actually be an open set, except {0}. Please help out.
Best Answer
This is a mistake in the book. As you correctly found, for the metric $$d^{\ast\ast}(x,y) = \lvert x_1 - y_1\rvert + \lvert x_2 - y_2\rvert$$ the open balls are open squares (with the diagonals parallel to the coordinate axes). This metric induces the standard topology on $\mathbb{R}^2$, hence no nonempty finite set is open. Common names for this metric are $\ell^1$-metric (since it is induced by the $\ell^1$-norm), Manhattan metric (since Manhattan is somewhat famous for a more or less rectangular road network, so the distance one has to travel between two points is the sum of the north-south distance and the east-west distance) or Taxicab metric.
They probably intended to give the metric as $$d^{\ast\ast}(x,y) = \begin{cases}\qquad 0 &\text{if } x = y \\ \lVert x\rVert + \lVert y\rVert &\text{if } x \neq y \end{cases}$$ where $\lVert\,\cdot\,\rVert$ is a norm on $\mathbb{R}^2$ (often the Euclidean, aka $\ell^2$, norm). This metric has the stated property, for $x \neq 0$ and $0 < r < \lVert x\rVert$ the open ball $B_r(x)$ is the singleton $\{x\}$.
This latter metric is - in my opinion unfittingly - also known as the British Rail metric, SNCF metric, or post office metric. These names are unfitting, because if two points lie on the same line, one doesn't need to travel via London or Paris respectively to reach one's destination, whereas the metric says one does have to.
A more fitting name, under which I first encountered this metric, but which unfortunately doesn't seem to have spread, is the metric of Gaul fishmongers. ("Das Meer? Was hat denn das Meer mit meinen Fischen zu tun?" My translation: "The sea? What has the sea to do with my fish?"; because he buys the fish for his shop in Paris rather than fishing in the sea right behind the village as Asterix suggested.)