Definition of a Solid Sphere

Given a metric space $X$ with distance function $d$ and a positive real number $ϵ\in {\mathbb{R}}_{>0}$, the solid sphere (or ball) or radius $ϵ$ and center $p\in X$ is the set defined by

$$S_{ϵ}(p)=\{x\in X:d(p,x)\le ϵ\}$$

In the case where $X$ is finite, the volume of the (solid) sphere $S_{ϵ}(p)$ is its number of elements $|S_{ϵ}(p)|=\#S_{ϵ}(p)$.

This doesn’t have to be an actual sphere, its just an area where the distance between two vertices is constant.