Definition:Taxicab Metric/Graphical Example

Definition

This diagram shows the open $\epsilon$-ball $\map {B_\epsilon} {A; d_1}$ of point $A$ in the $\struct {\R^2, d_1}$ metric space where $d_1$ is the taxicab metric.

Note that $\epsilon = \epsilon_1 + \epsilon_2$.

Neither the boundary lines nor the extreme points are actually part of the open $\epsilon$-ball.

Sources

• 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $2$: Metric Spaces: $\S 2$: Metric Spaces: Exercise $2$
• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $2$: Continuity generalized: metric spaces: Exercise $2.6: 9$

Strictly speaking, the exercise specifically calls for $\map {B_\epsilon} {0; d_1}$ where $0 := \tuple {0, 0}$.