Euclidean Metric and Chebyshev Distance on Real Metric Space give rise to Same Topological Space

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 $3$