Let $\displaystyle X=\bf R^3$ and define the metric d: $\displaystyle \bf R^3 \times \bf R^3 \rightarrow \bf R$ by

$\displaystyle d((x_1,y_1,z_1),(x_2,y_2,z_2)) = \max\left\{|x_1-x_2|, |y_1-y_2|, |z_1-z_2|\right\}$.

Describe the neighborhood $\displaystyle N(0; 1)$, where $\displaystyle 0 $ is the origin in $\displaystyle \bf R^3$

My professor says that it's an open cube with vertices at (±1, ±1, ±1), but I can't see why it's not an open sphere with radius 1.

I'm thinking $\displaystyle N(0; 1) = \left\{y \in X: d(0,y) <1\right\}$. Wouldn't there be points whose distance from the origin exceed 1 in an open cube?