PlanetMath: when are balls separated

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when are balls separated

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

To prove this, suppose that and are not separated. Then there exists a $z\in X$ such that either

$\displaystyle d(x,z)<r, \quad d(y,z)\le s,$

$\displaystyle d(x,z)\le r, \quad d(y,z)< s.$

In either case,

$\displaystyle d(x,y)\le d(x,z)+d(z,y)<r+s.$

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Cross-references: separated, balls, radius, open ball, metric space
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This is version 3 of when are balls separated, born on 2005-05-17, modified 2007-03-21.
Object id is 7066, canonical name is WhenAreBallsSeparated.
Accessed 926 times total.

Classification:

AMS MSC:	54-00 (General topology :: General reference works )
	54D05 (General topology :: Fairly general properties :: Connected and locally connected spaces )

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