January 18, 2022

Globe in metric space

Let M be a metric space. A sphere (open) with radius r > 0 and centered at point p in M ​​is defined as REMOVE r ( p ) { x ∈ USA | d ( x , p ) < r } , {\displaystyle B_{r}(p)\{x\in M\mid d(x,p)0. A unit sphere (closed or open) is a sphere of radius r equal to either of the above definitions. A subset of a metric space is said to be bounded if it is contained in some sphere. A set is said to be completely bounded if, given any radius r, a finite number of spheres of radius r can be found that cover the set. Open spheres with metric d produce a base of the induced topology by d (by definition). This means that all open sets in a metric space can be represented by the union of some open sphere.

Euclidean Sphere

In n-dimensional Euclidean space, with the usual metric; if this space is a straight line then the open sphere is an interval; and if this space is a plane, then the open sphere is a disk inside the circle. A closed unit sphere is usually denoted by Dn; The outer part of this sphere is an n-1 sphere, denoted Sn-1. For example, the 3-dimensional sphere S3 would be the "outside" (or boundary) of D4. The two concepts of sphere and sphere in space with higher number of dimensions are often called supersphere and supersphere. More about the concepts of "volume" and "area" can be found in the case of a space with dimensions greater than 3. With different metrics, the shape of the sphere in the same space can be different. Eg: In 2D space: With norm-1 (i.e. in taxicab geometry), the sphere is a square whose diagonals are parallel to the coordinate axes. With standard induction from the Chebyshev distance, the sphere is a square whose sides are parallel to the coordinate axes. In 3D space: With norm-1, the sphere is a regular octahedron with body diagonals parallel to the coordinate axes. With the Chebyshev distance magnetic induction standard, the sphere is a cube whose sides are parallel to the coordinate axes.

Globe in topological space

We can introduce the concept of a sphere in any topological space, without necessarily having it sensitive to some metric. A sphere (pick up

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