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.

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But how would I formalize this statement mathematically?


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


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