Open Ball in Cartesian Product under Chebyshev Distance
Jump to navigation
Jump to search
Theorem
Let $M_1 = \struct {A_1, d_1}, M_2 = \struct {A_2, d_2}, \ldots, M_n = \struct {A_n, d_n}$ be metric spaces.
Let $\ds \AA = \prod_{i \mathop = 1}^n A_i$ be the cartesian product of $A_1, A_2, \ldots, A_n$.
Let $d_\infty: \AA \times \AA \to \R$ be the Chebyshev distance on $\AA$:
- $\ds \map {d_\infty} {x, y} = \max_{i \mathop = 1}^n \set {\map {d_i} {x_i, y_i} }$
where $x = \tuple {x_1, x_2, \ldots, x_n}, y = \tuple {y_1, y_2, \ldots, y_n} \in \AA$.
Let $a = \tuple {a_1, a_2, \ldots, a_n} \in \AA$.
Let $\epsilon \in \R_{>0}$.
Let $\map {B_\epsilon} {a; d_\infty}$ be the open $\epsilon$-ball of $a$ in $M = \struct {\AA, d_\infty}$.
Then:
- $\ds \map {B_\epsilon} {a; d_\infty} = \prod_{i \mathop = 1}^n \map {B_\epsilon} {a_i; d_i}$
Proof
Let $\epsilon \in \R_{>0}$.
Let $x = \tuple {x_1, x_2, \ldots, x_n} \in \AA$.
Then:
\(\ds x\) | \(\in\) | \(\ds \map {B_\epsilon} {a; d_\infty}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map {d_\infty} {x, a}\) | \(<\) | \(\ds \epsilon\) | Definition of Open $\epsilon$-Ball | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \max_{i \mathop = 1}^n \set {\map {d_i} {x_i, a_i} }\) | \(<\) | \(\ds \epsilon\) | Definition of Chebyshev Distance | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \forall i \in \set {1, 2, \ldots, n}: \, \) | \(\ds \map {d_i} {x_i, a_i}\) | \(<\) | \(\ds \epsilon\) | Definition of Maximum Element | |||||||||
\(\ds \leadsto \ \ \) | \(\ds \forall i \in \set {1, 2, \ldots, n}: \, \) | \(\ds x_i\) | \(\in\) | \(\ds \map {B_\epsilon} {a_i; d_i}\) | Definition of Open $\epsilon$-Ball | |||||||||
\(\ds \leadsto \ \ \) | \(\ds x\) | \(\in\) | \(\ds \prod_{i \mathop = 1}^n \map {B_\epsilon} {a_i; d_i}\) | Definition of Finite Cartesian Product | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map {B_\epsilon} {a; d_\infty}\) | \(\subseteq\) | \(\ds \prod_{i \mathop = 1}^n \map {B_\epsilon} {a_i; d_i}\) | Definition of Subset |
And then:
\(\ds x\) | \(\in\) | \(\ds \prod_{i \mathop = 1}^n \map {B_\epsilon} {a_i; d_i}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \forall i \in \set {1, 2, \ldots, n}: \, \) | \(\ds x_i\) | \(\in\) | \(\ds \map {B_\epsilon} {a_i; d_i}\) | Definition of Finite Cartesian Product | |||||||||
\(\ds \leadsto \ \ \) | \(\ds \forall i \in \set {1, 2, \ldots, n}: \, \) | \(\ds \map {d_i} {x_i, a_i}\) | \(<\) | \(\ds \epsilon\) | Definition of Open $\epsilon$-Ball | |||||||||
\(\ds \leadsto \ \ \) | \(\ds \max_{i \mathop = 1}^n \set {\map {d_i} {x_i, a_i} }\) | \(<\) | \(\ds \epsilon\) | Definition of Maximum Element | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map {d_\infty} {x, a}\) | \(<\) | \(\ds \epsilon\) | Definition of Chebyshev Distance | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds x\) | \(\in\) | \(\ds \map {B_\epsilon} {a; d_\infty}\) | Definition of Open $\epsilon$-Ball | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \prod_{i \mathop = 1}^n \map {B_\epsilon} {a_i; d_i}\) | \(\subseteq\) | \(\ds \map {B_\epsilon} {a; d_\infty}\) | Definition of Subset |
The result follows by definition of set equality.
$\blacksquare$
Sources
- 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $2$: Metric Spaces: $\S 4$: Open Balls and Neighborhoods: Exercise $7$