# If $A$ is an open set of the taxicab metric $(R^2,d_1)$ then $A$ is a union of open balls of the Euclidian $(R^2,d)$ metric.)$Is is true that if$A$is an open set of the taxicab metric$(R^2,d_1)$then$A$is a union of open balls of the Euclidian$(R^2,d)$metric. Where as taxicab metric I mean$R^2$equipped with the distance$d((x_1,y_1),(x_2,y_2)) = |x_1 - x_2| + |y_1 - y_2| $. Intuitively I think it is possible to "fill" an open set of the taxicab metric (a square) with open sets of the Euclidian metric (circles) in such a way that the area of the gaps goes to zero by repeatedly inserting smaller an smaller balls in the gaps. But how would I formalize this statement mathematically? ## 1 Answer Note that you don't need a disjoint union of open ball. So you can do it like this : $$]-1,1[^2 = \bigcup_{(x,y) \in ]-1,1[^2} B\left( (x,y), \frac{1}{2} ( 1- \max(|x|,|y|) ) \right)$$ Explanation :$B\left( (x,y), \frac{1}{2} ( 1- \max(|x|,|y|) ) \right) $is the open ball centered on$(x,y)$with radius$\frac{1}{2} ( 1- \max(|x|,|y|) ) \$. And it's not hard to see that this ball is strictly inclued in the square.

But every point in the open square is the center of a ball inclued in the open square, so we just take the union of such balls