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)<r,\quad d(y,z)\leq s,

d(x,z)\leq r,\quad d(y,z)<s.

In either case,

d(x,y)\leq d(x,z)+d(z,y)<r+s.

Title	when are balls separated
Canonical name	WhenAreBallsSeparated
Date of creation	2013-03-22 15:16:40
Last modified on	2013-03-22 15:16:40
Owner	matte (1858)
Last modified by	matte (1858)
Numerical id	6
Author	matte (1858)
Entry type	Example
Classification	msc 54-00
Classification	msc 54D05