# Definition:Separated Sets

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## Definition

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

Let $A, B \subseteq S$.

### Definition 1

$A$ and $B$ are **separated (in $T$)** if and only if:

- $A^- \cap B = A \cap B^- = \O$

where:

### Definition 2

$A$ and $B$ are **separated (in $T$)** if and only if there exist $U,V\in\tau$ with:

- $A \subset U$ and $U \cap B = \O$
- $B \subset V$ and $V \cap A = \O$

where $\O$ denotes the empty set.

$A$ and $B$ are said to be **separated sets (of $T$)**.

## Also known as

When $A$ and $B$ are **separated in $T$**, they are said **to separate $T$**.

## Also see

- Equivalence of Definitions of Separated Sets
- Results about
**separated sets**can be found here.