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

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

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


 * $T$ admits a separation


 * there exist two closed sets of $T$ which form a (set) partition of $S$.
 * 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 $S$ $\relcomp S A \mid \relcomp S B$ is a (set) partition of $S$.

Hence the result by definition of closed set.