Countability Axioms Preserved under Open Continuous Surjection

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

Let $\phi: T_A \to T_B$ be a surjective open mapping which is also continuous.

If $T_A$ has one of the following properties, then $T_B$ has the same property:


 * First-Countability
 * Second-Countability