Definition 1

Given a point $x_0 \in \mathbf{R}^n$ and a real number $r > 0$, the open (resp. closed) Euclidean ball of $n$ dimensions with centre $x_0$ and radius $r$ is the set of all points $x \in \mathbf{R}^n$ such that $d(x_0, x) < r$ [resp. $d(x_0, x) \leq r$]; the Euclidean sphere of $n - 1$ dimensions with centre $x_0$ and radius $r$ is the set of all $x \in \mathbf{R}^n$ such that $d(x_0, x) = r$.

When there is no risk of confusion we say simply "ball" (resp. "sphere") for "Euclidean ball" (resp. "Euclidean sphere"). When $n = 2$, a ball of two dimensions is called a disc, and a sphere of one dimension is called a circle. When $n = 1$, the open (resp. closed) ball with centre $x_0$ and radius $r$ is the interval $]x_0 - r, x_0 + r[$ (resp. $[x_0 - r, x_0 + r]$); the sphere with centre $x_0$ and radius $r$ is the set consisting of the two end-points $x_0 - r, x_0 + r$ of these intervals.

From what has been said, the balls (open or closed) with centre $x_0$ (or just those with radii $1/p$, where $p$ runs through the set of integers $> 0$) form a fundamental system of neighbourhoods of the point $x_0$.