Separated sets

Two sets in a topological space that do not meet each other's closure.

Separated sets

Two separated sets $A$ and $B$ in a topological space $X$ are subsets such that

A\cap \overline{B}=\varnothing \quad\text{and}\quad \overline{A}\cap B=\varnothing,

where $\overline{A}$ and $\overline{B}$ denote closures in $X$ .

Separatedness is the key notion used to define connectedness : a space is disconnected exactly when it can be written as a union of two nonempty separated sets.

Examples:

In $\mathbb{R}$ with the usual topology, $A=(0,1)$ and $B=(1,2)$ are separated.
In $\mathbb{R}$ , the rationals $\mathbb{Q}$ and the irrationals $\mathbb{R}\setminus\mathbb{Q}$ are not separated, since each is dense and has closure $\mathbb{R}$ .