Second-Countability is not Continuous Invariant

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

Let $\phi: T_A \to T_B$ be a continuous mapping.

If $T_A$ is a second-countable space, then it does not necessarily follow that $T_B$ is also second-countable.

Proof
Let $T_S = \left({S, \tau_S}\right)$ be the Arens-Fort space.

Let $T_D = \left({S, \tau_D}\right)$ be the discrete space, also on $S$.

As $S$ is countable, from Arens-Fort Space is Expansion of Countable Fort Space, it follows that $T_D = \left({S, \tau_D}\right)$ is a countable discrete space.

Let $I_S: S \to S$ be the identity mapping on $S$.

From Mapping from Discrete Space is Continuous, we have that $I_S$ is a continuous mapping.

Then we have that a Countable Discrete Space is Second-Countable.

We have that the Arens-Fort Space is not First-Countable.

It follows from Second-Countable Space is First-Countable that the Arens-Fort space is not second-countable either.

Thus we have demonstrated a continuous mapping from a second-countable space to a space which is not second-countable.