Definition:Separated Sets

Definition

Let $T = \left({S, \tau}\right)$ be a topological space.

Let $A, B \subseteq S$.

$A$ 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.

$A$ 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 $T$).

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