# when are balls separated

Let $(X,d)$ be a metric space, and let $B_{r}(x)$ be the $x$-centered open ball of radius $r$. If $d(x,y)\geq r+s$, then the balls $B_{r}(x)$ and $B_{s}(y)$ are separated.

To prove this, suppose that $B_{r}(x)$ and $B_{s}(y)$ are not separated. Then there exists a $z\in X$ such that either

 $d(x,z)

or

 $d(x,z)\leq r,\quad d(y,z)

In either case,

 $d(x,y)\leq d(x,z)+d(z,y)
Title when are balls separated WhenAreBallsSeparated 2013-03-22 15:16:40 2013-03-22 15:16:40 matte (1858) matte (1858) 6 matte (1858) Example msc 54-00 msc 54D05