# Homeomorphism Relation is Equivalence

## Theorem

Let $T_1$ and $T_2$ be topological spaces.

Let $T_1 \sim T_2$ denote that $T_1$ and $T_2$ are homeomorphic.

The relation $\sim$ is an equivalence relation.

## Proof

Checking in turn each of the criteria for equivalence:

### Reflexivity

Let $T$ be a topological space.

From Identity Mapping is Homeomorphism, the identity mapping $I_T: T \to T$ is a homeomorphism.

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

$\Box$

### Symmetry

Let $T_1$ and $T_2$ be topological spaces such that $T_1 \sim T_2$.

By definition, there exists a homeomorphism $f: T_1 \to T_2$.

From Inverse of Homeomorphism is Homeomorphism it follows that $f^{-1}: T_2 \to T_1$ is also a homeomorphism.

So $T_2 \sim T_1$, and $\sim$ has been shown to be symmetric.

$\Box$

### Transitivity

Let $T_1, T_2, T_3$ be topological spaces such that $T_1 \sim T_2$ and $T_2 \sim T_3$.

By definition, there exist homeomorphisms $f: T_1 \to T_2$ and $g: T_2 \to T_3$.

From Composite of Homeomorphisms is Homeomorphism it follows that $g \circ f: T_1 \to T_3$ is also a homeomorphism.

So $T_1 \sim T_3$, and $\sim$ has been shown to be transitive.

$\Box$

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

Hence by definition it is an equivalence relation.

$\blacksquare$

## Sources

- 1964: Steven A. Gaal:
*Point Set Topology*... (previous) ... (next): Chapter $\text {I}$: Topological Spaces: $1$. Open Sets and Closed Sets - 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $3$: Continuity generalized: topological spaces: $3.6$: Homeomorphisms: Definition $3.6.1$ - 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $3$: Continuity generalized: topological spaces: Exercise $3.9: 16$