Isometry is Homeomorphism of Induced Topologies

Theorem
Let $\left({S_1,d_1}\right)$ and $\left({S_2,d_2}\right)$ be metric spaces.

Let $f \colon S_1 \to S_2$ be an isometry from $\left({S_1,d_1}\right)$ to $\left({S_2,d_2}\right)$.

Let $\tau_1$ and $\tau_2$ be the topologies induced on $S_1$ and $S_2$ by the metrics $d_1$ and $d_2$, respectively.

Then $f$ is a homeomorphism from $\left({S_1,\tau_1}\right)$ to $\left({S_2,\tau_2}\right)$

Proof
By the definition of an isometry, $f$ is bijective.

By Continuous Mapping on Topological Space Induced by Metric, $f$ is a continuous mapping from $\left({S_1,\tau_1}\right)$ to $\left({S_2,\tau_2}\right)$.

By Inverse of Isometry is Isometry, $f^{-1}$ is an isometry.

By Continuous Mapping on Topological Space Induced by Metric once more, $f^{-1}$ is a continuous mapping from $\left({S_2,\tau_2}\right)$ to $\left({S_1,\tau_1}\right)$.

Thus $f$ is a homeomorphism from $\left({S_1,\tau_1}\right)$ to $\left({S_2,\tau_2}\right)$.