Projection from Metric Space Product with Taxicab Metric is Continuous/Proof 1
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Theorem
Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be metric spaces.
Let $\AA := A_1 \times A_2$ be the cartesian product of $A_1$ and $A_2$.
Let $\MM = \struct {\AA, d}$ denote the metric space on $\AA$ where $d: \AA \to \R$ is the taxicab metric on $\AA$:
- $\map d {x, y} := \map {d_1} {x_1, y_1} + \map {d_2} {x_2, y_2}$
where $x = \tuple {x_1, x_2}, y = \tuple {y_1, y_2} \in \AA$.
Let $\pr_1: \MM \to M_1$ and $\pr_2: \MM \to M_2$ denote the first projection and second projection respectively on $\MM$.
Then $\pr_1$ and $\pr_2$ are both ā€ˇcontinuous on $\MM$.
Proof
The taxicab metric is an instance of the $p$-product metric:
- $\map {d_p} {x, y} := \paren {\paren {\map {d_1} {x_1, y_1} }^p + \paren {\map {d_2} {x_2, y_2} }^p}^{1/p}$
where $p = 1$.
The result is therefore seen to be an instance of Projection from Metric Space Product with P-Product Metric is Continuous.
$\blacksquare$