separated

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

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 $ℝ$, 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 $B_s(y)$ areseparated (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

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