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

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 $Y$ .

Examples

1. On , 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.).
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

Bibliography

1

J.L. Kelley, General Topology, D. van Nostrand Company, Inc., 1955.

2

G.J. Jameson, Topology and Normed Spaces, Chapman and Hall, 1974.