sphere (metric space)

The set $\{x\mid d(x,c)=r\}$ is called the sphere of radius $r$ with centre $c$ . This generalizes the notion of spheres to metric spaces.

Note that the sphere in a metric space need not look like a sphere in Euclidean space. For instance, if we impose the metric $d(x,y)=max\{|x_{1}-y_{1}|,|x_{2}-y_{2}|,|x_{3}-y_{3}|\}$ on $\mathbb{R}^{3}$ instead of the Euclidean metric, spheres according to this metric are actually cubes! Even more bizarre situations can occur in general — a sphere might be disconnected, or it may be discrete, or it may even be an empty set.

Title	sphere (metric space)
Canonical name	SpheremetricSpace
Date of creation	2013-03-22 14:47:38
Last modified on	2013-03-22 14:47:38
Owner	rspuzio (6075)
Last modified by	rspuzio (6075)
Numerical id	6
Author	rspuzio (6075)
Entry type	Definition
Classification	msc 54E35
Synonym	sphere