Sphere (metric sphere)

The set of points at distance exactly r from a center point in a metric space.

Sphere (metric sphere)

Let $(X,d)$ be a metric space, let $x\in X$ , and let $r\ge 0$ . The (metric) sphere of radius $r$ centered at $x$ is

S(x,r):=\{y\in X : d(x,y)=r\}.

Spheres generalize the usual circles and spheres in Euclidean geometry. They are useful for describing boundaries of balls and for constructing examples in metric topology.

Examples:

In $\mathbb{R}$ , $S(a,r)=\{a-r,a+r\}$ if $r>0$ .
In $\mathbb{R}^2$ , $S(0,1)=\{(x,y):x^2+y^2=1\}$ is the unit circle.
In a discrete metric space, $S(x,1)=X\setminus\{x\}$ .