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Homesphere (metric space)

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# 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.

Synonym:

sphere

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Definition

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## Mathematics Subject Classification

54E35*no label found*

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