Definition:Separated Sets

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:

$A^-$ denotes the closure of $A$ in $T$

$\O$ 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 $U \cap B = \O$

$B \subset V$ 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.