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 $T = \struct {S, \tau}$ 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
- 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 $\relcomp S A \mid \relcomp S B$ is a (set) partition of $S$.
Hence the result by definition of closed set.
$\blacksquare$