Definition:Open Ball/P-adic Numbers
< Definition:Open Ball(Redirected from Definition:Open Ball in 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
- 2007: Svetlana Katok: p-adic Analysis Compared with Real: $\S 2.1$ Elementary topological properties