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.
随便看	orientation Orientation1 OrientedCobordism oriented cobordism Origin origin OriginsOfWedderburnsTheorem origins of Wedderburn’s theorem OrnsteinUhlenbeckProcess Ornstein-Uhlenbeck process OrnsteinWeissLemma Ornstein-Weiss lemma OrthicTriangle orthic triangle Orthocenter orthocenter OrthocomplementedLattice orthocomplemented lattice Orthogonal orthogonal OrthogonalCircles orthogonal circles OrthogonalDecompositionTheorem orthogonal decomposition theorem OrthogonalDirectSum