# 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. 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 $Y$.

## Examples

1. 1.

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

2. 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. 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.

## Remarks

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

## 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 Separated 2013-03-22 15:16:34 2013-03-22 15:16:34 matte (1858) matte (1858) 15 matte (1858) Definition msc 54-00 msc 54D05