Homeomorphic Metric Spaces are not necessarily Isometric

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 homeomorphic.

Then it is not necessarily the case that $M_1$ and $M_2$ are isometric.

Proof
Consider the spaces $\struct {\openint 0 4, d}$ and $\struct {\openint 0 1, d}$, where $d$ is the Euclidean Metric.

By Open Real Intervals are Homeomorphic, they are homeomorphic.

Let $\phi: \openint 0 4 \to \openint 0 1$ be a mapping.

Then:

showing that $\phi$ cannot be an isometry.

Therefore the two spaces above are not isometric.