Isometry between Metric Spaces is Continuous/Corollary

Corollary to Isometry between Metric Spaces is Continuous

Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be metric spaces.

Let $\phi: M_1 \to M_2$ be an isometry.

Then its inverse $\phi^{-1}: M_2 \to M_1$ is a continuous mapping.

Proof

From Inverse of Isometry of Metric Spaces is Isometry, $\phi^{-1}$ is an isometry.

The result follows from Isometry between Metric Spaces is Continuous.

$\blacksquare$

Sources

• 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $2$: Metric Spaces: $\S 7$: Subspaces and Equivalence of Metric Spaces: Lemma $7.5$