Completely Hausdorff Space is Preserved under Homeomorphism

Theorem
Let $T_A = \left({X_A, \vartheta_A}\right), T_B = \left({X_B, \vartheta_B}\right)$ 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
From the equivalent definitions of homeomorphism, $\phi$ is a closed bijection.

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