# 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) SpheremetricSpace 2013-03-22 14:47:38 2013-03-22 14:47:38 rspuzio (6075) rspuzio (6075) 6 rspuzio (6075) Definition msc 54E35 sphere