Connected iff no Proper Clopen Sets

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Theorem

Let $T = \struct {S, \tau}$ 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 $\O \ne H \ne S$.

Then $H$ and $\relcomp S H$ are open sets whose union is $S$.

Thus $\set {H \mid \relcomp S H}$ 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 $\set {A \mid B}$ of $S$.

Then $\relcomp S A = B$ is open, and $A$ is closed.

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

Finally, $T$ must be connected.

$\blacksquare$