Connected iff no Proper Clopen Sets

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Theorem

Let $T = \left({S, \tau}\right)$ 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 $\complement_S \left({H}\right)$ are open sets whose union is $S$.
Thus $\left[{H \mid \complement_S \left({H}\right)}\right]$ 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 $\complement_S \left({A}\right) = B$ 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.

$\blacksquare$