Condition on Connectedness by Clopen Sets

From ProofWiki
Jump to navigation Jump to search

Theorem

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


Then:

$T$ admits no separation

if and only if:

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


Sources