separated
Definition Suppose $A$ and $B$ are subsets of a topological space^{} $X$. Then $A$ and $B$ are separated provided that
$$\begin{array}{ccc}\hfill \overline{A}\cap B\hfill & \hfill =\hfill & \hfill \mathrm{\varnothing},\hfill \\ \hfill A\cap \overline{B}\hfill & \hfill =\hfill & \hfill \mathrm{\varnothing},\hfill \end{array}$$ 
where $\overline{A}$ is the closure operator^{} (http://planetmath.org/Closure^{}) in $X$.
Properties

1.
If $A,B$ are separated in $X$, and $f:X\to Y$ is a homeomorphism^{}, then $f(A)$ and $f(B)$ are separated in $Y$.
Examples

1.
On $\mathbb{R}$, the intervals $(0,1)$ and $(1,2)$ are separated.

2.
If $d(x,y)\ge r+s$, then the open balls^{} ${B}_{r}(x)$ and ${B}_{s}(y)$ are separated (proof.) (http://planetmath.org/WhenAreBallsSeparated).

3.
If $A$ is a clopen set, then $A$ and ${A}^{\mathrm{\complement}}$ are separated. This follows since $\overline{S}=S$ when $S$ is a closed set^{}.
Remarks
References
 1 J.L. Kelley, General Topology, D. van Nostrand Company, Inc., 1955.
 2 G.J. Jameson, Topology and Normed Spaces, Chapman and Hall, 1974.
Title  separated 

Canonical name  Separated 
Date of creation  20130322 15:16:34 
Last modified on  20130322 15:16:34 
Owner  matte (1858) 
Last modified by  matte (1858) 
Numerical id  15 
Author  matte (1858) 
Entry type  Definition 
Classification  msc 5400 
Classification  msc 54D05 