# Definition:Separated Sets

Jump to navigation
Jump to search

## Definition

Let $T = \left({S, \tau}\right)$ 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 $A^-$ denotes the closure of $A$ in $T$, and $\O$ denotes the empty set.

### Definition 2

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

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

where $\varnothing$ 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.