Connected iff no Proper Clopen Sets

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

Then $T$ is connected if and only if there exists no proper subset of $S$ which is clopen in $T$.

Proof
Assume that $T$ is connected.

Let $H \subset S$ be a clopen set such that $\varnothing \ne H \ne S$.


 * Then $H$ and $\complement_S(H)$ are open sets whose union is $S$.


 * Thus $[H \mid \complement_S(H)]$ form a partition of $S$.


 * By definition, $T$ is not connected, which is a contradiction.


 * Then such $H$ does not exist.

Assume that there exists no proper subset of $S$ which is clopen in $T$.

Suppose $T$ is not connected (disconnected).


 * By definition, there is a partition $[A \mid B]$ of $S$.


 * Then $\complement_S(A) = B$ is open, and $A$ is closed.


 * Thus $A$ is clopen and $\varnothing \ne A \ne S$, which is a contradiction.


 * Finally, $T$ must be connected.