# Separated Sets are Disjoint

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## 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 |

$\blacksquare$

## Sources

- 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**separated sets**