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.

单词	SpheremetricSpace
释义	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.
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