Taxicab Metric on Metric Space Product is Continuous

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Theorem

Let $M = \struct {A, d}$ be a metric space.

Let $\AA$ be the Cartesian product $A \times A$.

Let $d_1$ be the taxicab metric on $\AA$:

$\ds \map {d_1} {x, y} = \sum_{i \mathop = 1}^n \map {d_{i'} } {x_{i'}, y_{i'} }$

for $x = \tuple {x_1, x_2, \ldots, x_n}, y = \tuple {y_1, y_2, \ldots, y_n} \in \AA$.


Then $d_1: \AA \to \AA$ is a continuous function.


Proof

Recall the definition of continuous mapping in this context.

Given metric spaces $M_X = \struct {X, d_X}$ and $M_Y = \struct {Y, d_Y}$, and a mapping $f : X \to Y$, we say that $f$ is $\struct {X, d_X} \to \struct {Y, d_Y}$-continuous if and only if:

$\forall x_0 \in X: \forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall x \in X: \map {d_X} {x, x_0} < \delta \implies \map d {\map f x, \map f {x_0} } < \epsilon$

Hence it is necessary to prove that $d: \struct {A \times A, d_1} \to \struct {\R, \size {\,\cdot \,} }$ is continuous.

That is, given $\tuple {x_0, y_0} \in A \times A$ and $\epsilon>0$, it is necessary to find $\delta \in \R_{>0}$ such that:

$\forall x, y \in A: \map {d_1} {\tuple {x, y}, \tuple {x_0, y_0} } < \delta \implies \size {\map d {x, y} - \map d {x_0, y_0} } < \epsilon$


Hence, let $\tuple {x_0, y_0} \in A \times A$.

If $\map {d_1} {\tuple {x, y}, \tuple {x_0, y_0} } = \map d {x, x_0} + \map d {y, y_0} < \epsilon$, then:

$\size {\map d {x, y} - \map d {x_0, y_0} } \le \size {\map d {x, y} - \map d {x, y_0} } + \size {\map d {x, y_0} - \map d {x_0, y_0} } \le 2 \epsilon$

using Metric Space Axiom $(\text M 2)$: Triangle Inequality.

The result follows.

$\blacksquare$


Sources

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