# 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 $\varnothing \ne H \ne S$.

- 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.

$\blacksquare$