Equivalence of Definitions of Connected Topological Space/No Union of Separated Sets implies No Continuous Surjection to Discrete Two-Point Space

Theorem
Let $T = \left({S, \tau}\right)$ be a topological space.

Let $T$ be such that there are no two non-empty separated sets whose union is $S$.

Then there exists no continuous surjection from $T$ onto a discrete two-point space.

Proof
Let $D = \left({\left\{{0, 1}\right\}, \tau}\right)$ be the discrete two-point space on $\left\{{0, 1}\right\}$.

Suppose $f: T \to \left\{{0, 1}\right\}$ is a continuous mapping.

Then $f^{-1} \left({0}\right)$ and $f^{-1} \left({1}\right)$ are open sets of $T$.

From the definition of a mapping:
 * $f^{-1} \left({0}\right) \cup f^{-1} \left({1}\right) = S$

and
 * $f^{-1} \left({0}\right) \cap f^{-1} \left({1}\right) = \varnothing$

Then:
 * $f^{-1} \left({0}\right) = S \setminus f^{-1} \left({1}\right)$

and:
 * $f^{-1} \left({1}\right) = T \setminus f^{-1} \left({0}\right)$

are clopen.

From Closed Set Equals its Closure they are their respective closures.

It follows from the definition that $f^{-1} \left({0}\right)$ and $f^{-1} \left({1}\right)$ are separated subsets of $T$ whose union is $S$.

Hence, by hypothesis, one of them must be empty, and the other one must be $S$.

Therefore $f$ is constant, and so is not a surjection.

That is, there exists no continuous surjection from $T$ onto a discrete two-point space.