Hausdorff Condition is Preserved under Homeomorphism

Theorem
Let $T_A = \left({S_A, \tau_A}\right), T_B = \left({S_B, \tau_B}\right)$ be topological spaces.

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

If $T_A$ is a $T_2$ (Hausdorff) space, then so is $T_B$.

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

The result follows from $T_2$ (Hausdorff) Space is Preserved under Closed Bijection.