Equivalence Class of Isometries is Subset of Equivalence Class of Homeomorphisms

Theorem
Let $\mathcal C_I$ be an equivalence class of isometries on the set of metric spaces.

Then $\mathcal C_I$ is a subset of an equivalence class of homeomorphisms on the set of metric spaces.

Proof
From Isometry of Metric Spaces is Equivalence Relation, all isometries can be partitioned into equivalence classes.

From Homeomorphism of Metric Spaces is Equivalence Relation, all homeomorphisms can be partitioned into equivalence classes.

Let $\mathcal C_I$ be an equivalence class of isometries.

Let $f: M_1 \to M_2$ be an element of $\mathcal C_I$ where $M_1$ and $M_2$ are some metric spaces.

From Isometry of Metric Spaces is Homeomorphism, $\mathcal C_I$ is a homeomorphism.

Thus $\mathcal C_I$ is an element of some equivalence class of homeomorphisms on the set of metric spaces.