Hausdorff Condition is Preserved under Homeomorphism

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

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

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