Continuous Image of Separable Space is Separable

Definition
Let $T_1 = \left({X_1, \tau_1}\right), T_2 = \left({X_2, \tau_2}\right)$ be topological spaces.

Let $f: T_1 \to T_2$ be a continuous mapping.

If $T_1$ is separable, then so is $T_2$.

Proof
From the definition of separable, $T_1 = \left({X_1, \tau_1}\right)$ is separable if there exists a countable subset of $X_1$ which is everywhere dense.

We need to show that if there exists a mapping $f: T_1 \to T_2$ which is continuous, then $T_2$ is also separable.

That is, there exists a countable subset of $X_2$ which is everywhere dense.