Definition:Closed Ball/P-adic Numbers

Definition
Let $p$ be a prime number.

Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers with $p$-adic norm $\norm {\,\cdot\,}_p : \Q_p \times \Q_p \to \R_{\ge 0}$.

Let $a \in \Q_p$.

Let $\epsilon \in \R_{>0}$ be a strictly positive real number.

The closed $\epsilon$-ball of $a$ in $\struct {\Q_p, \norm {\,\cdot\,}_p }$ is defined as:


 * $\map { {B_\epsilon}^-} a = \set {x \in R: \norm {x - a}_p \le \epsilon}$

Note
By definition, the $p$-adic numbers are the unique (up to isometric isomorphism) non-Archimedean valued field that completes $\struct {\Q, \norm {\,\cdot\,}_p}$ and $\norm {\,\cdot\,}_p$ is a non-Archimedean norm.

The definition of a closed $\epsilon$-ball of $a$ in $\struct {\Q_p, \norm {\,\cdot\,}_p }$ is nothing more than a specific instance of the general definition of a closed ball in a normed division ring.

Also see

 * Leigh.Samphier/Sandbox/Definition:Open Ball in P-adic Numbers
 * Leigh.Samphier/Sandbox/Definition:Sphere in P-adic Numbers