Equivalence of Definitions of Connected Topological Space/No Separation iff No Clopen Sets

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Theorem

The following definitions of the concept of Connected Topological Space are equivalent:

Let $T = \struct {S, \tau}$ be a topological space.

Definition by Separation

$T$ is connected if and only if it admits no separation.

Definition by No Clopen Sets

$T$ is connected if and only if its only clopen sets are $S$ and $\O$.


Proof

Definition by No Clopen Sets implies Definition by Separation

Let $T$ be connected by having no clopen sets.

Aiming for a contradiction, suppose $T$ admits a separation, $A \mid B$ say.

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

From this contradiction it follows that $T$ can admit no separation

$\Box$


Definition by Separation implies Definition by No Clopen Sets

Let $T$ be connected by admitting no separation.

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

Then $\relcomp S H$ is also clopen.

Hence $H \mid \relcomp S H$ is a separation of $T$.

From this contradiction it follows that $T$ can have no non-empty proper subsets which are clopen.

$\blacksquare$


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