# Separated Sets are Disjoint

## Theorem

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

Let $A, B \subseteq S$ such that $A$ and $B$ are separated in $T$.

Then $A$ and $B$ are disjoint:

$A \cap B = \O$

## Proof

Let $A$ and $B$ be separated in $T$.

Then:

 $\displaystyle A^- \cap B$ $=$ $\displaystyle \O$ Definition of Separated Sets: $A^-$ is the closure of $A$ $\displaystyle \leadsto \ \$ $\displaystyle \paren {A \cup A'} \cap B$ $=$ $\displaystyle \O$ Definition of Set Closure: $A'$ is the derived set of $A$ $\displaystyle \leadsto \ \$ $\displaystyle \paren {A \cap B} \cup \paren {A' \cap B}$ $=$ $\displaystyle \O$ Intersection Distributes over Union $\displaystyle \leadsto \ \$ $\displaystyle A \cap B$ $=$ $\displaystyle \O$ Union is Empty iff Sets are Empty

$\blacksquare$