Equivalence of Definitions of Connected Topological Space/No Clopen Sets implies No Union of Separated Sets

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

Let the only clopen sets of $T$ be $S$ and $\varnothing$.

Then there are no two non-empty separated sets of $T$ whose union is $S$.

Proof
Suppose $A$ and $B$ are separated subsets of $T$ such that $A \cup B = S$.

Then $A \cap B^- = \varnothing$ and $S = A \cup B \subseteq A \cup B^- \subseteq S$.

Hence $A = S \setminus B^-$.

From Topological Closure is Closed, $B^-$ is closed in $T$.

Thus $A$ is open in $T$.

By the same reasoning, $B$ must also be open.

But $A \cap B \subseteq A \cap B^- = \varnothing$, and $A \cup B = S$, by assumption.

So $A = S \setminus B$ and $B = S \setminus A$, and we conclude that both $A$ and $B$ are clopen.

Therefore, by hypothesis, one of them must be $S$ and the other must be $\varnothing$.

That is, there are no two non-empty separated sets of $T$ whose union is $S$.