Continuous Image of Connected Space is Connected

Theorem
Let $T_1$ and $T_2$ be topological spaces.

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

If $T_1$ is connected, then so is $f \left({T_1}\right)$.

Proof
By Continuity of Composite with Inclusion: Inclusion on Mapping, the surjective restriction $f_1: T_1 \to f \left({T_1}\right)$ of $f$ is continuous.

So it is enough to prove the result where $f: T_1 \to T_2$ is a surjection.

So, in this case, suppose $A \mid B$ is a partition of $T_2$.

Then it follows that $f^{-1} \left({A}\right) \mid f^{-1} \left({B}\right)$ is a partition of $T_1$.

Hence the result, by the Rule of Transposition.