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 non-empty proper subset of $S$ which is clopen in $T$.

Proof

Sufficient Condition

Let $T$ be connected.

Aiming for a contradiction, suppose there exists $H \subset S$ such that:

$H$ is clopen in $T$

$H$ is a non-empty proper subset of $S$, that is:

$\O \ne H \ne S$

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

Thus $\set {H \mid \relcomp S H}$ form a partition of $S$.

Hence by definition, $T$ is not connected.

This contradicts our assumption.

Hence, by Proof by Contradiction, such $H$ does not exist.

$\Box$

Necessary Condition

Let there exist no non-empty proper subset of $S$ which is clopen in $T$.

Aiming for a contradiction, 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.

Hence by definition $A$ is closed.

Thus $A$ is clopen such that:

$\O \ne A \ne S$

But then by definition $A$ is a non-empty proper subset of $S$ which is clopen in $T$.

This contradicts our assumption.

Hence, by Proof by Contradiction, $T$ is connected.

$\blacksquare$