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[parent] sphere (metric space) (Definition)

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 \{\vert x_1-y_1\vert, \vert x_2-y_2\vert, \vert x_3-y_3\vert\}$ 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.



"sphere (metric space)" is owned by rspuzio.
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Other names:  sphere

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Cross-references: empty set, discrete, disconnected, occur in, even, cubes, Euclidean metric, metric, Euclidean space, metric spaces, centre, radius
There are 33 references to this entry.

This is version 3 of sphere (metric space), born on 2004-11-04, modified 2005-05-02.
Object id is 6446, canonical name is SphereMetricSpace.
Accessed 4849 times total.

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AMS MSC: 54E35 (General topology :: Spaces with richer structures :: Metric spaces, metrizability)
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