# 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 $\varnothing$.

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

## Proof

### 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 $\varnothing$.

$\Box$

### 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.

$\blacksquare$

## Also see

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

## Sources

- 1964: Steven A. Gaal:
*Point Set Topology*... (previous) ... (next): $\S 1.1$ - 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $6.2$: Connectedness: Corollary $6.2.4$