Definition:Open Ball/P-adic Numbers

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Definition

Let $p$ be a prime number.

Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.

Let $a \in R$.

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


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

$\map {B_\epsilon} a = \set {x \in \Q_p: \norm{x - a}_p < \epsilon}$


Radius

In $\map {B_\epsilon} a$, the value $\epsilon$ is referred to as the radius of the open $\epsilon$-ball.


Center

In $\map {B_\epsilon} a$, the value $a$ is referred to as the center of the open $\epsilon$-ball.


Also see


Sources