Isometry is Homeomorphism of Induced Topologies

Theorem

Let $\struct {S_1, d_1}$ and $\struct {S_2, d_2}$ be metric spaces or pseudometric spaces.

Let $f: S_1 \to S_2$ be an isometry from $\struct {S_1, d_1}$ to $\struct {S_2, d_2}$.

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 $\struct {S_1, \tau_1}$ to $\struct {S_2, \tau_2}$.

Proof

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

By Continuous Mapping is Continuous on Induced Topological Spaces, $f$ is a continuous mapping from $\struct {S_1, \tau_1}$ to $\struct {S_2, \tau_2}$.

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

By Continuous Mapping is Continuous on Induced Topological Spaces once more, $f^{-1}$ is a continuous mapping from $\struct {S_2, \tau_2}$ to $\struct {S_1, \tau_1}$.

Thus $f$ is a homeomorphism from $\struct {S_1, \tau_1}$ to $\struct {S_2, \tau_2}$.

$\blacksquare$