Equivalence of Definitions of Connected Topological Space/No Separation iff No Union of Closed Sets

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Theorem

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

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

Definition by Separation

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

Definition by Union of Disjoint Closed Sets

$T$ is connected if and only if it has no two disjoint nonempty closed sets whose union is $S$.


Proof

From Biconditional Equivalent to Biconditional of Negations it follows that the statement can be expressed as:

$T$ admits a separation

if and only if:

there exist two closed sets of $T$ which form a (set) partition of $S$.


By definition, a separation of $T$ is a (set) partition of $S$ by $A, B$ which are open in $T$.


From Complements of Components of Two-Component Partition form Partition:

$A \mid B$ is a (set) partition of $S$ if and only if $\complement_S \left({A}\right) \mid \complement_S \left({B}\right)$ is a (set) partition of $S$.

Hence the result by definition of closed set.

$\blacksquare$