Completely Normal Space is Preserved under Homeomorphism

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

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

If $T_A$ is a completely normal space, then so is $T_B$.

Proof
We have that $\left({X, \tau}\right)$ is a completely normal space iff:


 * $\left({X, \tau}\right)$ is a $T_5$ space
 * $\left({X, \tau}\right)$ is a $T_1$ (Fréchet) space.

From $T_5$ Space is Preserved under Homeomorphism:
 * If $T_A$ is a $T_5$ space, then so is $T_B$.

From $T_1$ Space is Preserved under Homeomorphism:
 * If $T_A$ is a $T_1$ (Fréchet) space, then so is $T_B$.

Hence the result.