Euclidean Metric and Chebyshev Distance on Real Metric Space give rise to Same Topological Space
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Theorem
For $n \in \N$, let $\R^n$ be an Euclidean space.
Let $d_2$ be the Euclidean metric on $\R^n$.
Let $d_\infty$ be the Chebyshev distance on $\R^n$.
Let $T_2 = \struct {\R^n, \tau_2}$ denote the topological space which is induced by $d_2$.
Let $T_\infty = \struct {\R^n, \tau_\infty}$ denote the topological space which is induced by $d_\infty$.
Then $T_2$ and $T_\infty$ are the same.
That is:
- $\tau_2 = \tau_\infty$
Proof
From P-Product Metrics on Real Vector Space are Topologically Equivalent, $\tau_2$ and $\tau_\infty$ are topologically equivalent metrics.
The result follows from Topologically Equivalent Metrics induce Equal Topologies.
$\blacksquare$
Sources
- 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $3$: Topological Spaces: $\S 2$: Topological Spaces: Exercise $3$