# Completely Hausdorff Space is Preserved under Homeomorphism

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## Theorem

Let $T_A = \struct {S_A, \tau_A}$ and $T_B = \struct {S_B, \tau_B}$ be topological spaces.

Let $\phi: T_A \to T_B$ be a homeomorphism.

If $T_A$ is a $T_{2 \frac 1 2}$ (completely Hausdorff) space, then so is $T_B$.

## Proof

By definition of homeomorphism, $\phi$ is a closed continuous bijection.

The result follows from $T_{2 \frac 1 2}$ (Completely Hausdorff) Space is Preserved under Closed Bijection.

$\blacksquare$

## Sources

- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $2$: Separation Axioms: Functions, Products, and Subspaces