Isometric Metric Spaces are Homeomorphic

Theorem

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

Let $M_1$ and $M_2$ be isometric.

Then $M_1$ and $M_2$ are homeomorphic.

Proof

By the definition of an isometry, there exists a bijection $f: A_1 \to A_2$ such that:

$\forall a, b \in A_1: \map {d_1} {a, b} = \map {d_2} {\map \phi a, \map \phi b}$

From Isometry of Metric Spaces is Homeomorphism, $f$ is a homeomorphism from $M_1$ to $M_2$.

The result follows by definition of homeomorphic metric spaces.

$\blacksquare$

Sources

• 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $2$: Metric Spaces: $\S 7$: Subspaces and Equivalence of Metric Spaces: Corollary $7.7$