Completely Hausdorff Space is Preserved under Homeomorphism

Theorem
Let $T_A = \struct {S_A, \tau_A}, T_B = \struct {S_B, \tau_B}$ 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 a homeomorphism, $\phi$ is a fortiori a closed bijection.

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