separated

Definition Suppose $A$ and $B$ are subsets of a topological space $X$ . Then $A$ and $B$ are separated provided that

\begin{array}[]{ccc}\overline{A}\cap B&=&\emptyset,\\ A\cap\overline{B}&=&\emptyset,\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 $f\colon X\to Y$ is a homeomorphism, then $f(A)$ and $f(B)$ are separated in $Y$ .

1.

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

If $d(x,y)\geq r+s$ , 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 $A^{\complement}$ are separated. This follows since $\overline{S}=S$ when $S$ is a closed set.

The above definition follows [1]. In [2], separated sets are called strongly disjoint sets.