Homeomorphism of Metric Spaces is Equivalence Relation

Theorem
Let $M_1$ and $M_2$ be metric spaces.

Let $M_1 \sim M_2$ denote that $M_1$ and $M_2$ are homeomorphic.

The relation $\sim$ is an equivalence relation.

Proof
Checking in turn each of the criteria for equivalence:

Reflexivity
Let $M$ be a metric space.

From Identity Mapping on Metric Space is Homeomorphism, the identity mapping $I_M: M \to M$ is a homeomorphism.

So $M \sim M$, and $\sim$ has been shown to be reflexive.

Symmetry
Let $M_1$ and $M_2$ be topological spaces such that $M_1 \sim M_2$.

By definition, there exists a homeomorphism $f: M_1 \to M_2$.

From Inverse of Homeomorphism between Metric Spaces is Homeomorphism it follows that $f^{-1}: M_2 \to M_1$ is also a homeomorphism.

So $M_2 \sim M_1$, and $\sim$ has been shown to be symmetric.

Transitivity
Let $M_1, M_2, M_3$ be metric spaces such that $M_1 \sim M_2$ and $M_2 \sim M_3$.

By definition, there exist homeomorphisms $f: M_1 \to M_2$ and $g: M_2 \to M_3$.

From Composite of Homeomorphisms between Metric Spaces is Homeomorphism it follows that $g \circ f: M_1 \to M_3$ is also a homeomorphism.

So $M_1 \sim M_3$, and $\sim$ has been shown to be transitive.

$\sim$ has been shown to be reflexive, symmetric and transitive.

Hence by definition it is an equivalence relation.