Condition on Connectedness by Clopen Sets

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

Then:
 * $T$ admits no separation


 * the only clopen sets of $T$ are $S$ and $\varnothing$.
 * the only clopen sets of $T$ are $S$ and $\varnothing$.

Thus both conditions can be used to define a connected topological space.

Necessary Condition
Then by definition $T$ admits a separation, $A \mid B$ say.

Then both $A$ and $B$ are clopen sets of $T$, neither of which is either $S$ or $\varnothing$.

Sufficient Condition
Suppose $\exists H \subseteq S$ which is clopen.

Then $\complement_S \left({H}\right)$ is also clopen and so $H \mid \complement_S \left({H}\right)$ is a separation of $T$.

Hence by definition, $T$ is not connected.

The result follows by definition of connectedness.

Also see

 * Connected Topological Space: Definition from Separation
 * Connected Topological Space: Definition from Clopen Sets