Condition on Connectedness by Clopen Sets

Theorem
Let $$T$$ be a topological space.

Then $$T$$ is connected iff the only clopen sets of $$T$$ are $$T$$ and $$\varnothing$$.

Proof
By definition of connectedness, $$T$$ is connected iff it admits no partition.


 * Let $$T$$ be not connected.

Then by definition $$T$$ admits a partition, $$A | B$$ say.

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


 * Now suppose $$\exists A \subseteq T$$ which is clopen.

Then $$T - A$$ is also clopen and so $$A | T - A$$ partitions $$T$$.

Hence by definition, $$T$$ is not connected.

The result follows by definition of connectedness.